Well, the default option is for an email notification on new conversations. So most people would get it.
For anyone else: If a warning puts them over the ban threshold they will see when they next try to log in "You have been banned for: Rules violation" which is the generic message. Else, if you've banned them directly through the Admin CP as long as you've provided a reason they will see "You have been banned for: Whatever reason you provided".
Not sure why it's necessary to definitely notify them by e-mail. If they get the e-mail as a result of the private chat, fine. But if they don't, so what. They'll soon know they've been banned.