This is referring specifically to chat, as opposed to posting, as we have guidelines over scope and each site can determine if a topic is right for that site. Chat is amorphous to some extent, although the Be Nice policy does extend into all areas of the site.
There's a recent experience, I don't want to say where or with whom, but suffice to say on the Stack Exchange Network with a user in chat.
The user announced that they did something illegal for a living. Spamming people with phone calls for fake software help to get credit card info (something along the lines of such).
People in the chat room didn't like this and we voiced our opinions. Turns out the user will still bring it up and sulk about it some time later. This got me thinking.
How should we respond to people who announce in chat they are engaged in illegal work?