If you're arrested, do the cops have the automatic right to video while you're being questioned and show that video on TV?
However, there are two main defenses to a claim of defamation. The first is the absolute defense of truth. If the statement made is true, then the suit for defamation will fail. The second is consent. If you consent to a statement being published about you, and the statement is published within the scope of that consent, then you cannot later maintain a suit for defamation.
Both defenses would apply to the broadcasters of the show Cops. Where a television show depicts a person being arrested, and that person was in fact arrested, then the depiction is true. Further, I understand that Cops places a disclaimer at the beginning of the show stating that the people depicted in the show are innocent until proven guilty in a court of law. This statement more specifically clarifies for the viewer that the person depicted did not necessarily commit the crime, but had merely been arrested for it.
As to consent, I understand that Cops often obtains release forms from the people being depicted. This release form constitutes consent, and so the person depicted has no legal recourse for defamation so long as the television station acts within the scope of the release.
Also, anyone depicted on the show whose face is blurred out is unlikely to win a defamation suit. This is because the tort of defamation also requires that a person watching the show can identify the subject.
Another related defense to defamation is newsworthiness. For example, if a suspect is driving erratically because he is drunk, and the video of the chase is later played as a news story about the dangers of driving under the influence, then the television station can invoke a defense of newsworthiness.