Senior policy adviser Stephen Miller talks to reporters about President Donald Trump's support for creating a ”merit-based immigration system.” (Chip Somodevilla/Getty Images)

Reporters are supposed to push their sources for answers. But how far can they push before “aggressive reporting” starts to look like badgering, grandstanding or just plain rude?

Wednesday’s White House briefing provided Exhibits A and B for debating that question.

Reporters Jim Acosta of CNN and Glenn Thrush of the New York Times took on Stephen Miller, the White House senior policy adviser, in exchanges that were memorable for the heat, if not the light, they generated. The run-ins spurred debate about where the line between legitimate journalistic inquiry ends and incivility begins.

Thrush first challenged Miller about a White House-backed legislative proposal that would slash the number of legal immigrants. He repeatedly asked Miller for statistics to support Miller’s contention that low-skilled immigrant workers displace American workers and depress wages.

Both men interrupted each other repeatedly, until Miller finally snapped and snipped. “Glenn, maybe we’ll make a carve-out in the bill that says the New York Times can hire all the low-skilled, less-paid workers they want from other countries and see how you feel then about low-wage substitution,” he said.

CNN Senior White House Correspondent Jim Acosta in the briefing room Wednesday. (Chip Somodevilla/Getty Images)

And that was just the warm-up act. Acosta then engaged Miller on a facet of the proposal that would give priority to immigrants who speak English. He cited the Emma Lazarus poem enshrined on the Statue of Liberty (“Give me your poor, your tired, your huddled masses yearning to breathe free . . . ”) and asked whether the plan was discriminatory. “Are we just going to bring in people from Great Britain and Australia?” he said.

Miller responded by saying the question revealed Acosta’s “cosmopolitan bias.” “Jim, have you honestly never met an immigrant from another country who speaks English, outside of Great Britain and Australia?” Miller asked. “Is that your personal experience?”

Acosta’s aggressive questioning of Trump administration officials has made him a polarizing figure; former White House press secretary Sean Spicer temporarily banned live TV coverage of the briefings largely in response to him. Thrush, meanwhile, has become something of a folk hero; he’s the “Glenn” featured in Melissa McCarthy’s sendup of Spicer on “Saturday Night Live.”

Their latest exchanges predictably created a divided reaction. Some razzed Miller. Others, however, found the reporters out of bounds, particularly Acosta. In a tweet, Ari Fleischer, George W. Bush’s press secretary, said, “@acosta an advocate for a political point of view, not as a neutral reporter. He’s not even playing the devil’s advocate. This is bias.

Both Acosta and Thrush say they have no regrets for aggressively seeking answers.

“I was pressing the guy for a statistical justification for a policy that will affect millions of Americans, and he decided to deliver a speech,” says Thrush, via email. “The studies he cited are well-known and have been interpreted in a lot of different ways. I wanted to see if he’d acknowledge that ambiguity.” He added, “If you aren’t stopping a politician from blah-blah-blah-ing, you’re not doing your job.”

Acosta was similarly unmoved: “As my mother told me recently, ‘Let other people be the wallflower,’ ” he said in a brief interview. “If quoting from the Statue of Liberty is pushing too hard, I’m going to keep pushing.”

Acosta offered several more takes on CNN, opining later Wednesday that “I think what you saw unfold in the briefing room is that he [Miller] really just couldn’t take that kind of heat and exploded before our eyes.” And: “It’s not often you’re accused of a cosmopolitan bias from someone who went to Duke University wearing cuff links in the White House briefing room.”

A request for comment from Miller was not immediately returned.

To some extent, the testy exchanges are remarkable only because viewers can see them. It’s not uncommon for reporters to rile their sources with persistent, even rude, questions, though they tend to do so with few witnesses. News briefings carried on live TV expose this messy aspect of reporting — showing, in effect, how the news sausage is sometimes made. As Thrush puts it, “I wish the soaps were still around so people would watch something else to get their midday entertainment.”

Edward Wasserman, dean of UC-Berkeley’s journalism school, says viewers were well served by the Acosta-Miller throwdown. “If [Acosta] had been more timid and deferential, those of us who watched it would have felt underserved and underrepresented,” he said. “It certainly had a bruising and edgy element, and there was no love lost, but at the same time, it felt illuminating.”

He added: “This is the way you provoke a response that exposes the thinking behind policy. . . . People aren’t normally privy to [the reporting process], but that’s what a journalist is supposed to do.”

But Lucy Dalglish, who heads the University of Maryland’s journalism school, confessed to feeling “unsettled” by watching a reporter and a source go at it.

“There’s something to be said for civility,” she said. “Good reporters don’t want to make themselves the story. It’s also true that good public servants don’t want to make this all about them.”

Some of American journalism’s giants, she said, showed that it was possible to be both insistent and persistent without being abrasive or sparking an argument. Her shortlist: Sam Donaldson, Ted Koppel and Dan Rather.

“I like allowing people to finish a sentence,” Dalglish said, referencing both sides of the briefing-room transaction. “I got annoyed not as much about an attitude but because I wanted to hear the end of a thought.”