The city of Rockville has a new way of mowing its grass and collecting litter: It hires private citizens for the job and saves $8,000 a year. As part of a recent productivity campaign, Maryland's second largest city also earned $16,000 by increasing newspaper recycling in the community and saved $750,000 by dropping liability insurance and covering claims itself.

"Looking for ways to stretch tax dollars to the max is happening everywhere," said Assistant City Manager Daniel Hobbs, who notes proudly that Rockville will be the host of a conference on its money-saving campaign next month for Maryland mayors and city administrators.

Rockville and other jurisdictions in the Washington area and around the country have been forced to develop some ingenious methods to lessen the impact of federal budget cuts and inflation. The result is hundreds of new, often unorthodox ways to cut costs and locate new resources, from a "Phantom Storm Drain Cleaner Campaign" in Baltimore to an "adopt-a-park" program in Wichita, Kan.

In some places, these efforts have been motivated by voter-mandated tax limits and by President Reagan's appeal for greater voluntarism. Almost everywhere, however, finding new ways to manage on less money has become an obsession for local officials, the subject of an endless number of seminars, books and newsletters.

Although everyone agrees these efforts cannot begin to make up the difference in billion-dollar federal budget cuts, a host of innovative methods are helping to keep cities, town and individuals afloat.

Typically, communities are seeking donated labor and materials, instituting volunteer efforts such as self-reading water meter programs, imposing new user fees for services like back yard trash collection, and consolidating police, fire, court and economic development services with neighboring towns.

Bartering, such as one town trading ambulance service for access to another town's sewage treatment plant, also is flourishing in the face of declining revenues.

"The time has come when we can no longer afford to fight turf battles with surrounding government bodies," said Robert J. Paciocco, former county administrator for Prince Edward County, Va.

In Prince George's County, Giant Food Inc. and 15 other employers agreed to provide job training to 50 welfare mothers at no cost to the county as part of "Project Independence," a program established by the county to help find jobs for welfare recipients.

In addition, traditional volunteer programs at local firehouses, libraries, schools and police stations have been strengthened and a list of the county's thriftiest ideas, such as its just-approved plan to merge the county and school print shops at an initial savings of $300,000, have been collected in a popular booklet, "101 Ways to Squeeze More Out of Your Local Government Dollar." Strict monitoring of sick leave and overtime in the last year also are expected to save $113,000 yearly, according to Major Riddick, principal budget analyst for the county.

Montgomery County was able to slash insurance costs, as was Rockville, by dropping all insurance coverage except for catastrophes. Both governments found it cheaper to cover losses themselves than to pay ever-growing insurance premiums.

"It certainly is the wave of the '80s, how to do more with less," said Suzanne Muncy, director of Montgomery County's Office of Management and Budget human services division. This year, county officials persuaded private lawyers to donate time to make up the federal cuts in legal services; local businesses joined with the county in training 288 working women whose welfare payments were reduced, and the Marriott Corp. is funding an antidrug program for teen-agers. Another plan, which sought to supplement $363,000 in county money for groceries for the poor with donations from supermarkets, failed to get the necessary support from grocers.

In the District, Control Data Corp. gave a $118,000 computer system to the public schools and IBM and Xerox provided money for summer jobs programs. Geico Insurance Company has donated money and expertise to untangle the city's voter lists, and Pepco lent several of its executives to help the city improve its billing systems.

Even the local jurisdictions that have not been hit hard by budget cuts are looking to citizens and businesses for help in trimming costs. Arlington County, for example, has saved nearly $10,000 this year by hiring 27 retirees and Boy Scout troops to maintain the grass on median strips throughout the county. "We're going to try to expand the list to small parks," said Park Director Fred Lewis, who noted that the citizen effort enabled to county to keep its maintenance crew at five employes.

Additional ways to increase business and volunteer contributions to government are being examined by a citizens committee, according to budget director Jim Maloney.

Baltimore, already a national leader in volunteer efforts, published a "wish list" in June of hundreds of items that the federal budget cuts put out of reach. From buying a $12 thesaurus for legal services for the elderly and new equipment for city playgrounds to providing firehouses with a new coat of paint, almost half of the wishes have been granted by the city's citizens.

City officials have succeeded in enlisting private help for a wide variety of other programs as well.

Last March, the city persuaded 33 companies to clean 415 storm drains for free in a two-day "Phantom Storm Drain Cleaners" campaign.

"We've moved from the traditional to involving people who have never been volunteers before," according to Fontaine Sullivan, an assistant to Mayor William Donald Schaefer, who said 50 cities have asked Baltimore how to set up a wish list.

The Mayor's Office of Manpower Resources is also placing unemployed youths in a school for paperhangers run by the Plymouth Wallpaper Company. The school was started by company owner Dorothy Swimmer in response to cuts in federal job training programs. "We can't train the thousands of people that the CETA program did," said George Kopp, manager of Plymouth's wallpaper school, "but we can do something."

Other firms and individuals share that attitude--with some unusual benefits for the city. WBAL, a Baltimore radio station, for example, paid $34,000 to buy the New York diner used in the movie, "Diner," and the city, with the help of private donations, will transform it into a teaching restaurant for high school vocational students.

In Howard County, private businesses have taken another approach. In response to a $100,000 challenge grant from the county, 100 firms have pledged to increase their charitable gifts to 5 percent of their pretax profits to help overloaded social service agencies.

A similar effort in Henrico County, Va., Businesses Who Care, is raising money from firms to make up some of the recent cuts in government services. The groups are among several dozen nationwide, often called 2 percent or 5 percent clubs, that have been formed in the last year to urge member firms to give a set percentage of pretax profits to charities.

Several national organizations, representing mayors, counties and townships, have set up clearinghouses in the last year to collect examples of cost-cutting measures and private and public partnerships. Many of the money-saving ideas revolve around recyling resources, making buildings more energy-efficient and reusing old facilities, but some are offbeat, like Falls City, Ore., population 820, where the City Council is selling potholes for $10 each to earn money to patch them. ("Deluxe models" cost $20 and the mayor spray paints the donor's name across the patch.)

"There's a significant increase in the number of partnership arrangements," said Scott Fosler, a Montgomery County councilman and director of government studies for the Committee for Economic Development, a private research group. "You get the gimmicky to those that show the fundamental way government is shifting to local resources."

But no matter how cost-conscious local governments become, or how many businesses donate money for library services, the "private sector doesn't have the financial resources to fill the gap," said Arthur G. Will, director of the Institute for Local Self-Government, a Berkeley, Calif., group working with counties and cities coping with tax limits imposed by voters.

Although corporate giving increased by $300 million, to $2.9 billion in 1981, the amount is a minor part, only 5.4 percent, of all charitable contributions in the United States. (The American Association of Fund-Raising Councils and The Conference Board, which monitor charitable giving, show that private citizens give about 85 percent of all charity dollars.) For 1982, "We know that (corporate) contributions are going up, but not dramatically because of lower profits," said Kay Troy of the Conference Board.

And when government joins the traditional community groups in seeking money from business, someone loses out. "There's a thin line we don't want to cross," said Muncy, of the Montgomery County budget office. "In the past we had the philosophy that government shouldn't go to private industry because the nonprofits need to use them and the county government had a built-in funding source. We don't want to join the thousands of groups pouring into the Chamber of Commerce seeking help."

Some critics charge that the publicity over these new efforts diverts attention from the seriousness of the budget cuts. The President's Task Force on Public Sector Initiatives, for example, blankets newspapers throughout the country with examples from its computer bank of 2,000 successful partnerships. The stories, "The Brighter Side of Today's News" are sometimes used verbatim by smaller newspapers.

"The president never said business would pick up every social program cut in the federal budget," said Jay Moorhead, a White House aide on private sector initiatives. "This is an effort that should always be part of government and is not meant to be a fill-the-gap kind of thing."

Moorhead and state and local officials do agree that the move toward seeking help from volunteers and businesses is likely to be a permanent change. "This is not just tied to President Reagan," Moorhead noted. "It parallels a shift that's been growing for years."