An e-mail is not automatically generated, but on the Warn screen there is a Notification tab. This allows you to start a Conversation with the member. They would then get an e-mail as a result of that - as long as that option isn't turned off.
In the Admin CP you can pre-define notification options for each Warn category.
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.
If it's a temporary ban an admin would waqnt the experience to be as courteous and non hostile as possible in order to salvage the relationship. So just dumping them in a ban without notification would deepen the rift.
If the ban is absolute, or you don't seriously think the temp ban is going to turn them around (you're doing it because you need to be seen to be fair) then it wouldn't matter. But if you're doing a temp ban as a kind of training rebuke then it does matter.