When a moderator responds to a comment flag, they can either validate or dismiss it. Validating the flag will increase the flagger's "helpful" counter and delete the comment, without any kind of penalty or notification to the commenter.
For most comment flags, no further action is necessary. Assuming the comment is truly egregious, or the commenter has exhibited a repeating pattern of disruptive behaviour, the responding moderator may proceed to warn them via a private e-mail or suspend their account. Moderators are generally encouraged to warn on first offense and only suspend when a behaviour persists, but the decision is completely up to the responding moderator1.
This is the general approach on all Stack Exchange sites. And it's far from foolproof. I've been moderating two sites for some time now, and I can tell you that it's extremely easy to miss a pattern of abusive comments. We do have access to a full history of comments (including deleted ones), but unless we have a good reason to go look at it, we won't.
If you happen to spot a repeating behaviour that hasn't been handled properly, please consider using the custom flag to clearly explain what's going on. A flag message like:
Hey, this guy has been posting abusive comments for a while now, could you please take a closer look?
is probably all you need to do to help the responding moderator realize that there's something more there than a single abusive comment.
For the more technical stuff, see: How does comment voting and flagging work?
1 The other moderators and Stack Exchange are automatically notified of all private e-mails and suspensions. Mistakes are very easy to identify and fix.