I refer you to the commenting privilege page:
Comments are temporary "Post-It" notes left on a question or answer. They can be up-voted (but not down-voted) and flagged, but do not generate reputation. There's no revision history, and when they are deleted they're gone for good.
Comments aren't meant to stick around for longer than a few weeks. Ideally, all useful information would be in posts where it's supposed to be.
But that's not how it works. People put actual, valuable stuff in comments. This information can and should be moved to answers at some point, but that's not the point here.
The point is that people tend to do this:
You really should add that teaching your students using an upside-down schoolbus may cause them to dream about the Cold War. - aGuy
Added, thanks - postAuthorGuy
Comment exchanges like above are useless, redundant information and should be deleted. As a mod, you don't need to be running around hunting for these, but if it bugs someone enough that they flag it you can and should delete it.
And then there's the somewhat official policy on mSO:
The [obsolete comment threads] don't add anything other than a distraction. I am often filled with disappointment when I see what looks like an interesting comment thread only to find it's mainly bunch of correction please and thank you's. I wouldn't mind seeing mine disappear.
And as yet another mSO answer says:
That's exactly what the
obsolete flag is for. Flag away!
Comments are, by their very nature, transitional. Flagging helps keep
them under control and tidied up.
So please delete them when they're obviously obsolete and someone flags them. Please.
If they contain more information than what's found in the post, and it's a similar comment exchange, edit it into the post! Then you can delete the comments since they're no longer needed.