When a moderator flags a comment, the comment is instantly deleted. For a moderator, the only difference between using the “flag” and the “delete” button is that using the “flag” button asks for a deletion reason. (Is this deletion reason recorded anywhere?) This differs from posts, where moderator attention flags work normally when they are raised by moderators.
Moderators will generally avoid to deal with comments that concern them. While as a moderator I have no qualms deleting comments like “@Gilles thanks for your observation, I've updated my post accordingly”, I will not touch comments like “@Gilles your the stepchild of Hitler and Justin Bieber you're mother was is a hampster FOAD”. I leave such comments to be handled by another moderator.
If I'm a moderator, my only in-band recourse with a rude or non-constructive comment that's directed at me is to flag the post that the comment is attached to, which leaves a record in the post instead, which is not good if the author of the post is not the commenter. I can contact other moderators out-of-band (in chat), but this is inefficient and hard to keep track of.
Please let moderators leave a comment flag which can then be handled by another moderator.