Joseph Chung is one of the lucky ones, an immigrant who entered the United States illegally and has since been able to legalize his status in this country.

His story dramatizes many of the problems the Carter administration will encounter in attempting to implement a proposal that would combine penalties for employers of illegal immigrants with an amnesty for millions of those immigrants who have built their lives here even though they entered the country without proper documents.

It demonstrates the near-impossibility of determining how many illegal immigrants there are because of the case with which many illegal immigrants - regardless of their country of origin - blend into large cities. It shows how difficult it will be to seek out employers of illegals, particularly in minority communities that are suspicious of outsiders. Finally, Chung's experiences suggest that the success of any amnesty will depend on the ability of the government to convince illegal immigrants that the program is not simply a ruse for finding them and deporting them.

Six years ago, Chung (not his real name) took a job as a cook's assistant on a ship leaving Hong Kong. He jumped ship near Baltimore and made his way to New York's Chinatown, where he was expected by a man who had been a friend of his uncle in Hong Kong.

Out of respect for Chung's uncle and in anticipation of a $5,000 fee, the friend found the boy a job as a cook in a Chinatown restaurant and a bride who was an American citizen. The job allowed Chung to pay off the middleman's fee in just under four years. The chinese-American bride allowed him to apply for and gain residence in the United States. Marrying an American citizen is one of the surest ways for an immigrant to legalize his or her immigration status. "I Was Lucky"

OVER TEA in a dumpling house on Mott Street, the 26-year-old Chung explained why he entered the United States without going through lawful immigration procedures. "I was turned down when I tried to enter a university in Hong Kong - there were only two of them, and dozens of applicants for every place. My father is a tailor and I know that all I can do is work in his shop if I stay in Hong Kong. We know that in America, it is much easier to go to college.

"I visited the American consul in Hong Kong and was told I had no chance of getting an immigrant visa, at least not for years. My family has no relatives in the United States; everyone with relatives would come ahead of me. And we are not refugees from the mainland; the refugees come ahead of us, too.

"So I decided to come here any way I can. I was lucky. My father's brother did have a friend here and he sent a letter to that friend telling him I would be coming. My uncle's friend is an honorable man, although is may not seem so to an American. He told me right at the beginning what I would have to pay, so I knew I would have to pay $25 a week for at least four years."

Joseph Chung, was married six months after he arrived in New York. His wife, Anna, is 10 years older than he is. She agreed to the match because she wanted a husband. It took 18 months from the time of the wedding for Chung to legalize his status with the New York office of the Immigration and Naturalization Service (INS). By that time, the Chungs had a baby.

While he was living in New York illegally and working as a cook in a succession of restaurants, Chung had a fairly elaborate set of identification papers, including a Social Security card, a driver's license and a health insurance card. None of the papers was forged, because no proof of citizenship was required to obtain them.

As soon as Chung received the INS papers attesting to his legality, he applied and was admitted to the City College of New York, a school that played a vital role in the upward mobility of previous generations of immigrants. He graduates this June and has been accepted to medical school for next fall. His wife is a nurse and Chung plans to continue working as a cook in spite of the demanding medical school curriculum.

"My marriage was arranged," he says, "but it was not - what do the immigration agents call it? - a fraud. My wife was 29 and had no husband. I needed to find a way of staying here inside the law. We both got what we expected. We have a child, we have a life.

"I would have preferred to immigrate legally; I was terrified all the time I was living here outside the law. Do I consider that I was a criminal? No, no. Is it criminal to want to work? When I sigend onto that ship, I thought to myself, the worst that can happen to you is you will be sent back. You will be no worse off than if you didn't try to come to America. A Lack of Information

CHUNG BELONGS to a category of immigrants who have not received much attention in the current debate over illegal immigration: those who were not impoverished in their homelands but who left in search of the greater educational and social mobility afforded by American society.

Most of these illegal immigrants generally live in large cities in the Northeast or the Midwest. Jumping ship is an unusually adventurous way to enter the country illegally; most of the immigrants arrive on legal tourist or student visas and simply remain the United States after their permits have expired.

The vast majority of illegal immigrants do come from Mexico or Central America and enter the United States by crossing the Mexican border. Unlike Chung, most of the Mexican immigrants are impoverished. They come to feed themselves and their families, fleeing a country where an estimated 50 per cent of adult of men are unemployed.

Approximately 90 per cent of the illegal immigrants apprehended by the INS are Mexicans, but that does not mean that 9 out of 10 illegal immigrants in the country are Mexican. It is easier for the INS to apprehend illegal entrants in the Southwest border area. INS officials say the true percentage of illegals who are Mexicans is probably somewhere between 70 and 80 per cent.

The INS's shaky and unverified estimate of the national origins of illegals is one example of the lack of solid information that has hampered the attempts of the Carter administration to put together a legislative package.

"We know we're working on the basis of pretty thin information," says a Justice Department aide who has been involved in all of the planning sessions. "Illegals live underground outside the usual institutions of our society. We're proceeding on the basis of what we do know - that millions of these people are already here, that something has to be done to ease their fears and bring them within the protection of the law at the same time that we have to discourage employers from trying to hire new arrivals."

The basic problem is that no one really knows how many illegal immigrants are living in the United States. The estimate most frequently cited during tha past two years have ranged from 4 to 12 million. Gen. Leonard F. Chapman Jr., who has just been succeeded as INS commissioner by Carter appointee Leonel J. Castillo, sometimes estimated the number at 20 million. Chapman's estimates, which have been widely publicized, played an important role in setting the tone of the public debate about illegal immigration.

While Chapman was making his estimates, officials in the INS Division of Planning and Evaluation quietly acknowledged that there was little factual basis for the figures.

They began planning a "residential study" in which investigators would go into immigrant communities to try to arrive at a more accurate estimate of the illegal population. The study is scheduled to begin this August but most of the new Carter appointees, including Castillo, doubt its value. The Labor Department turned down a request for money to help finance the project.

"We see an operation like this an essentially worthless," says an aide to Labor Secretary P. Ray Marshall. "How is the INS going to count the illegals, by knocking on doors in Chinatown or the barrios? You'll see a stream of people going out the back door the minute a researcher knocks on the front door to say how-do-you-do."

The number of illegal immigrants matters becaust it is related to both aspects of the plan Carter will propose: Penalties for employers of illegals and the amnesty. The legislative package is expected to include civil - but not crinimal - sanctions against employers and an amnesty for illegal immigrants who have been in the country at least five years.

The dual approach is based partly on the realization that no bill with employer sanctions is likely to get through Congress without an amnesty and no amnesty will be enacted without employer penalties that might discourage the flow of new illegal immigrants.

The more illegal immigrants there are, the harder it will be to make employer sanctions meaningful. Castillo says he believes it would take a police force that boggles the mind to enforce such a law.

In Cabinet-level consultations, Labor Secretary Marshall has been the most forceful spokesman for the belief that the presence of illegals depresses wages even when no American workers are actualy displayed. Marshall's belief has been heavily influenced by his experiences in Texas, where the economic impact of illegal immigration across the Mexican border is much more visible than the presence ofillegals in Northern cities.

Although unions have pushed hard for curbs on employers of illegal immigrants, many of them also organize illegals in states where it is possible to do so. The negative impact of illegal immigration on wages in the Southwest is greatly intensified by the weak position of unions in that section of the country.

In New York, for example, legal and illegal immigrantns account for a high proportion of workers in garment factories. The International Ladies Garment Workers Union (LGWU) has members from both groups of immigrants and they all make union wages. Significantly, both Chinese and Hispanic immigrant workers report, that it is harder to find garment factory work in New York today than it was five years ago. The reason, ILGWU officials say, is that many garment factories have moved to Florida and Texas, where they operate with non-union workers and lower labor costs. Staying of Welfare

JOSEPH CHUNG'S job history, both as an illegal and as a legal immigrant, aillustrates the complexity of the relationship between immigration and the American labor market.

Chung's first job after his arrival in New York was in a small restaurant that employed both legal and illegal immigrants. He made much less than the minimum wage - about $80 for a 72-hour work week. After he had been in the United States six months - while he was still an illegal - he found a job in a better restaurant outside of Chinatown. He became a member of Local 69 of the AFL-CIO Hotel, Restaurant Employees and Bartenders Union. His union card served as another piece of identification to reduce his chances of apprehension by the INS.

Chung has worked in restaurants throughout his years as a student at City College; nearly all of his fellow employers are Hispanic of Chinese immigrants. Some are here legally and some illegally. All make the minimym wage. Chung is convinced that no Americans would take those jobs even though union members do earn minimum wages.

"The Welfare in New York pays just about as much as an assistant cook gets," he says. "Immigrants, no matter how they entered this country, are ashamed of being on welfare much more than Americans are."

Chung's statement about the attitude of immigrants toward welfare is supported by the few studies that have been conducted on the subject. In southern California - home for large numbers of illegal Mexican immigrants - reviews of welfare caseloads found a minuscule number of illegals on welfare.

In a report prepared by San Diego County's Human Resources Agency, a review of 9,1932 welfare cases showed only 10 illegal immigrants on welfare. A similar investigation of 14,000 cases by Los ANgeles County turned up 56 illegal immigrants. In New York City, welfare caseworkers say that both legal and illegal immigrants are afraid to report to welfares offices for fear of being identifieid and deported.

As Chung talked about the distaste of immigrants for welfare, three of his friends nodded in agreement. All had entered the United States illegally and all were members of the restaurant workers union. Of the four men, only Chung had been able to legalize hisstatus. Each of the three illegals had at least three pieces of identification.

The widespread availability of both forged and real identification for illegals is another reason it will be difficult to enforce sanctions against employers. The issue of how to enforce-employer penalties without tampering with civil liberties of both employers and employees has caused the greatest dispute within the administration. Attorney General Griffin Bell has strongly opposed any system that might lead to the use of a national identity card. Marshall originally wanted the law to be enforced through a "non counterfeitable" Social Security card, but both Bell and Hew Secretary Joseph A. Califano Jr. opposed the idea.

Marshall dropped the proposal when it was estimated that it would cost $500 million to adapt Social Security cards for such use. Bell is opposed to the idea of any universal ID such as those used in the Soviet Union and most Western European countries.

"This would mean a fundamental change in our ideas about civil liberties and privacy," says Terry Adamson, Bell's special assistant. "There's no secret about our position on this of Justice; we 're opposed to it on civil liberties grounds.

Administration aides are now talking about some form of "non-counterfeitable" card for non-citizens alone. In the immediate future, employers would probably have to rely on a combination of ID cards to prove that they had not knowingly hired illegal immigrants. Labor Department aides acknowledge that it is easy to acquire a set of false ID cards, but the administration seems to be relying on the idea of "voluntary compliance" by employers.

Critics of such a bill maintain it will be as unenforceable as Prohibition and will lead to discrimination against American citizens from many minority groups.

Donald G. Hohl, associate director of the U.S. Catholic Conference's Migration and Refugee Service, says the crux of the problem is who will be required to carry what ID. Hohl, one of the most knowledgeable immigration specialists in the country, has been consulted extensively on the administration proposals.

"If an employer is asked to differentiate between an illegal immigrant and a citizen, who is going to be asked for an ID? A white American or a Mexican-American? A white American or a Chinese-American? It's absolutely vital, this question of how to enforce the law without discriminating against American citizena from minority groups."

Everyone agrees that new legislation is neede to deal with illegal immigration, Hohl says, "but the worrisome thing is that we'll end up with a law that's just an empty bag. Unenforceable penalties for employers and an amnesty you don't have the means to carry out could be worse than nothing." "I Worked So Hard to Get Here"

AT THE DUMPLING HOUSE on Mott Street, Chung's three friends said they were overjoyed at the thought that there might be an amnesty for illegal immgrants but worried about how they would prove their eligibility.

One man who has been in the United States five years said an amnesty would mean "everything" to him because he would be able to go to school and look for a better job.

"But how can I prove I have been here five years?" he asked. "The man who the restaurant where I work could say how long I have been here, but he would never do that. He would be afraid he might get inntrouble.

Administration aides are talking about "absolute criteria" for amnesty so that no immigrant will be in doubt about his or her status. Assuming the government agencies could agree on those criteria, the reaction of the men on Mott Street suggests that the terms of the amnesty will be difficult ot communicate to millions of fearful immigrants who normally avoid all government programs.

Joseph Chung does not need an amnesty, because he was lucky enough to have had an honest and efficient "middleman " when he jumped ship six years ago. But he hopes there will be an amnesty, because he knows his friends will never have a chance for a education and a profession as long as they are living here illegally.

"I am going to become a doctor," he says. I would never have that possibility if I were still living in fear."

Chung is sensitive to the accusation that immigrants are taking jobs that might be filled by Americans. While he maintains that no American would want his job as a cook, he acknowledges that there are many Americans who would like to get into medical school.

"I worked so hard to get here" is his reply to critics of Immigration. "The language for me is still a struggle, even though I went to English-language school in Hong Kong. I wasn't lucky enough to be born here. How many Americans work seven nights a week while they go to school? And also, the medical program I will enter is to prepare me for a public health clinic. I would like to work with other Asian-American; there is great need for Chinese-speaking doctors. The opportunities you - ((KEY OFF)(KEYWORD)mericans take for granted are a miracle to me."