It depends on the seriousness of the crime in question. If I said I had exceeded the legal speed limit while driving, or if someone said they purchases illegal substances (something some people here seem to do regularly) or download a pirated movie, no one would care. I doubt anyone is going to confess to murder or being a child molester, so I don't think serious crimes that I would report are going to come up.
Plus, if someone is engaging in idle expression of anger ("I'm gonna kill that guy if I see him again!") it's probably nothing to worry about.
However, if you truly suspect a serious crime that will have consequences is going to be committed, or if someone is trying to involve you in a serious crime that will cause a lot of harm, I think I would have to do something. Basically, the question I'd ask is how much harm will come of my inaction and how much effect will my action have. You then have to decide what your conscience can live with.