As a moderator, when I come across a comment that contains an insult or offensive wording, I have three options: delete the entire comment, edit the comment, or leave it alone.
If there's little redeeming value to the rest of the comment, and its removal won't tear apart a conversation, I usually delete the comment. It's the easiest way to deal with rudeness or offensive material, and 80+% of comments flagged for rudeness are simply deleted.
In the rare cases where the insult or rudeness are somehow core to the conversation (this is almost always localized to discussions on Meta about someone's behavior), I'll leave the comment alone.
I am reluctant to edit the comments of anyone, because I am afraid of putting words in your mouth. However, I will edit a comment if there is redeeming value in it that makes me hesitant to delete, and there's a clean way to remove an insult or offensive content without altering the intent of the comment. For example, I frequently come across comments like this on Stack Overflow:
You're an idiot. The C specification states on page 123 that the behavior of the froop function is undefined under the following conditions: [conditions].
The first sentence can easily be removed without changing any of the wording in the remainder, preserving that useful information without the insult. I'm not altering someone's phrasing, just cutting out an irrelevant insult.
Again, almost all of the time we'll just delete any comment containing insults or offensive wording, but there are rare cases where I feel it is justified to edit comments to preserve content.
Comment handling is by nature quiet. You don't get notifications on deletion or edits of your comments largely to prevent fights over trivial matters from escalating. It's my experience that silently removing arguments and insults is extremely effective at stopping feuds before they begin. Providing notifications to remind people of arguments is a surefire way to cause them to blow up.