I'm curious why tabloids haven't been sued out of existence. I do recall Carol Burnett getting a bit of remuneration for the heartache they caused her some years back, but surely there can't be so much apathy that celebrities will permit almost anything to be said about their lives. Maybe it's a subtle form of blackmail: "At least if they say I'm in rehab, they aren't exposing my extramarital affairs."
There are good reasons celebrities encounter difficulty getting a libel case against the press to stick, but let's note at the outset that currently the real action is in privacy violation. Silicon Valley, as we've recently discovered, is innovating the hell out of this arena. You probably saw the news that the media organization Gawker declared bankruptcy after fighting a series of lawsuits secretly funded by Peter Thiel, a tech gazillionaire with a grudge, his goal no less than to put Gawker out of business. After the knockout punch, a privacy suit over a Hulk Hogan sex tape resulting in a $140-million judgement, observers fretted that Thiel had single-handedly opened up a new front against the free press: If you've got enough money, you don't need to prove libel or privacy violation in your own case (Thiel objected to being quasi-outed as gay in a 2007 Gawker piece). You just have to spend eight or nine years burying your nemesis in other people's cases until you find one with enough merit to put 'em out of their misery.
OK, so this might be a little breathless. Who knows? Maybe all the public opprobrium will shame these billionaires into behaving. (Ha ha.) But Thiel's covert tactics reflect the robust good health of press-protection laws in the United States. Thiel might not have won a privacy case, since his sexual orientation was already an open secret, and he couldn't have won a libel case because Gawker would've argued the piece they published was true. But even if it hadn't been, that hardly would've mattered. Thiel's a "public figure"—part of a special, less-protected class as far as libel law is concerned. Put plainly, if you're the editor of the National Enquirer, you can print significantly nastier stuff about somebody famous than you can about, say, the schoolteacher next door. Public figures are still at an advantage relative to "public officials"—i.e., elected representatives—but not as likely to succeed in a libel case as "limited-purpose public figures," folks who've been thrust temporarily into the public eye—witnesses to a high-profile murder, for instance.
In order to win a libel suit against a news organization, public figures have to demonstrate that the offending party acted with "actual malice," which is not a Tom Clancy novel but rather a standard set by the Supreme Court's 1964 decision in New York Times v. Sullivan: They must show that the defendant knew for sure that the offending information was false, and published it anyway. Mind-reading being notoriously difficult, this sets a pretty high bar.
Not that there haven't been conspicuous celebrity wins. As you mention, in 1976, Carol Burnett went after the Enquirer for implying she'd been obstreperously drunk in public, despite sources' reports to the contrary. (A distinction to keep in mind: we're talking here about tabloids that traffic in salacious gossip, rather than those that print obviously absurd "Miranda Lambert Impregnated by Aliens!"-type material; any celebrity who'd sue over that stuff needs their head examined.) Drawing upon her considerable resolve and resources, Burnett prevailed in court and finally settled. Her success launched what the Atlantic calls the "modern era of tabloid litigation," wherein other slighted celebs were emboldened to fight back.
In response the tabs lawyered up to the nines. House counsel now put their eyes on everything at the major tabloids; one former Enquirer staffer has said that two attorneys there look at each piece, and not in some toothless advisory capacity—if they say kill it, it gets killed. These aren't strip-mall sleazebags, either. For instance, it was David Kendall, one of Bill Clinton's lawyers during various '90s scandals, who had earlier green-lighted an Enquirer story headlined "Liberace's Secret Battle with AIDS." (I guess I'm not making a particularly strong case here that the guy's not a sleazebag. He is, however, a very well-regarded one.)
If an article that makes it to print does ruffle any serious feathers, the tabs' significant legal manpower gives them the option of dragging a case out forever; and by suing, celebrities expose themselves to the discovery process, during which—as you correctly suggest—they might be required to cough up personal information they'd really rather not.
Altogether, such factors make it both tedious and legally difficult for celebrities to win a media libel case. The Sullivan standard, by the way, means the U.S. is pretty much sui generis when it comes to libel law, at least until President Trump takes office. "I'm going to open up our libel laws so when they write purposely negative and horrible and false articles, we can sue them and win lots of money," promised the Donald. It seems like the short-finger jokes are really starting to get to him.
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