Confronting a thief in person is pretty dangerous (if the police weren't there); there's a non-zero chance of a fight, possibly involving a knife or gun. If you theoretically have a CCW and can be legally carrying a gun for self defense, it's still physically dangerous, and legally risky (a lot of legal/self defense advice is that if you're going to confront someone like a cheating spouse or whatever, you should not have even a legally owned gun with you, since if things escalate, it can get much worse.) Even a "righteous" self defense shoot in California is probably going to cost you $50-100k in legal -- absolutely worth it to save your life or the life of someone you care about, not really worth it for anything else, including breaking up strangers fighting on the street.
Much safer to just gather information and give it to the police.
I don't know how much risk I'd be willing to take for a $350 camera. (if it had a 5.0 f/1.0L or 1200mm, capture/torture/killing might be appropriate, though)