TL;DR> Obviously, Cascabel here is flagging comments, and others do that too. But not enough people are chipping in to help.
In addition to tchrist's answer, note that moderators are exception handlers. The community is charged with moderation; moderators — unless they step in to nip something in the bud — deal with what the community cannot decide on.
If a comment is no longer necessary, flag it. Yes, it will go into the moderators' flag queue; but another flag (or maybe two) from members of the community will deal with the comment and remove it from moderators. If particular users get too many flags on their comments then moderators might get to know of that.
However, please don't search out comments to flag, particularly from posts which are years old. Those are unlikely to be flagged by others and will simply languish for moderators to deal with. Flags like that are more likely to be dismissed as unhelpful — the comments have been around for ages without harm and you've just dumped a load of ancient stuff on moderators' laps.
Of course, everyone would benefit if those who debate answers in comments actually wrote their own answer that could be voted on, and those who answer in comments actually wrote a real answer. If that's not done, then any pearls of wisdom may well be lost forever when moderators simply clear their queue. The community can help here by actively flagging recent comments which don't follow the guidance: it doesn't take many flags for a comment to be removed automatically, and after a while people may get the idea that comments are intended for a certain purpose.
Providing an answer which might "normally" go in a comment is still providing an answer. It may not be entirely complete, and others are free to build their own answer on that foundation; but if you're going to provide an answer, any answer, then put it in the answer box. Even if it's only a foundation for others, if it's actually useful it's unlikely to be voted down (I hope).
It's also possible to copy an answer-in-comment to the answer box and provide an actual answer; and flag the comment as No Longer Required. It would be good practice to acknowledge the source, but whether or not to make it a Community Wiki answer is up to the person who bothers to write the answer. Again, serial answerers-in-comments might get the message.
As Mari-Lou says, sometimes a simple reminder to users to get off your lawn is suffice for two users who are having a discussion or seem to be bickering between themselves. By all means flag such exchanges: moderators may deal with them before the community accumulates enough flags, but flags on comments are counted.