The work I do—what I spend most of my day doing—is moderating comments. That is, I read the comments, decide which ones to highlight or "Feature," sometimes (not always) do a little basic editing triage, check the links, and post them.
When someone gets mad, or there's a dispute, or I need to "disallow" (not publish) a comment, I usually—not always—contact the commenter privately.
It's important to note that I can't contact people who don't leave an email address. If you've ever had a comment that you've written not appear on the site and you got nuthin' but silence from me, it might be because you failed to disclose your contact information and I can't get in touch with you.
To proceed. Sometimes, comment moderation is easy...