To take on your question literally:
While ordinary citizens or organizations have a legal right to privacy, "public figures" or organizations don't. So, for example, if a newspaper or TV station uses your image without your permission, and you can prove that doing so caused you damage, you can sue. But if you're John Gotti Jr. you're a public figure, and the law assumes that, as a public figure, people are interested in you and have the right to see your image without having the media constrained.

Mafia families may think of themselves as a "secret society," but their criminal activites have been exposed so often and for so long that they're in the public domain. It's inconceivble that a Don could bring the news media, or government prosecutors into court and say, "We're a secret society and you're causing us damage by exposing us to cameras, news stories, indictments, etc." It's not inconceivable that a Mafia guy who was exposed as such in a news story, could sue the news medium for "defamation of character," hoping to win damages but mainly to throw the media and government off of the track. But if he did take his case to court, he'd have to prove damage--which would expose him to defense interrogation and investigation. His own boss probably would whack him before he went to trial. And, as dt points out, a Mafia guy's beef with a civilian would be settled with baseball bats and bullets.

Movies and TV series are somewhat different. A documentary is like a news story. But entertainment, which can include fictional dialog and events, is different. Did you notice that in the movie "Goodfellas," Jimmy the Gent Burke is called "Jimmy Conway" and Paul Vario is called "Paulie Cicero"? I'm guessing here but the probable reason is that both were alive when the movie was made. Even though they were public figures and had been convicted of crimes like those shown in the film, the fact that actors played them and had them uttering dialog that they may not have said in real life could have opened the studio to a lawsuit by them or their families. Another possibility: considering that Henry Hill was the source for this material, the studio had good reason to take everything he told them with a barrel of salt.