Comment flags, much like comments themselves, are pretty subjective. When a comment is flagged the mods see this in the flag dashboard with the question it belongs to, but not in-situ with other comments - just the comment itself and the flag reason along side it (it doesn't even show who flagged it - comment flagging is mostly* anonymous, even to mods).
Because comments are second-class citizens compared to the real posts that's how they're treated as far as moderation goes too. If it's a blatantly offensive / spammy post, or if it cites a link that is since dead then it'll just get deleted as that's usually a no-brainer.
In order to judge whether it's a chatty / obsolete / not constructive we have to go into the question itself, read through all the comments and see if it fits in context with everything in there or not. If we just act on the flag itself then it could leave the rest of the comments making no sense depending on whether the deleted comment related to any other comments, or if other comments are responses to the flagged one.
As far as how I personally handle comment flags; if I don't see a big need to delete it then I'd probably just leave it as-is. They're usually not doing any harm to the site and they give the site a bit more of a community feel, and deleting the odd comment in a post might break up the flow of the rest of the comments, thereby making them all nonsensical. I even tend to leave comments alone that criticize the OP of the question (those sort of comments get flagged quite a bit) as if it's a valid criticism then I see it as a valid comment (even if they can be a bit sharp sometimes).
My advice for comment flagging - save it for the really obviously useless comments (like the "+1 great post!" variety, or anything that is just blatantly offensive) and just leave the rest of them there.
Finally; if you're getting comment flags declined then don't take it personally - we don't know who it is that flagged them (that's not the case for Question / Answer flags though).
* I say 'mostly', but we can see in a users profile what their flagging stats among other things (so we can check if people are misusing the privilege, standard mod stuff) but when a comment is flagged itself it doesn't say who flagged it. That only shows for questions/answers that get flagged.