Option 1: Try to talk it out, one-on-one.
I have a short anecdote to start with.
Two and a half years ago, I was a newly minted moderator on History and Science & Mathematics. We'd been out of private beta for maybe two weeks, and I'd had a couple disagreements with a high-rep user who was (and still is) one of our most valuable contributors. It wasn't too serious, but I was getting a little annoyed with some of his behavior, and I didn't want to screw up, as a mod leading a young, unsteady site.
We had a brief discussion in chat, over a couple days (time zone differences made it hard). I only briefly brought up our differences of opinion, and, if I recall correctly, we got to talking about, the history of math - one of the subjects the site was about, a topic we were both passionate about, something we could really connect with.
See, what I remembered then was that Stack Exchange is about content. We're about questions and answers. We might have differences of opinion, but we're all here because we care about a couple of subjects, and we want to make the Internet a better place. We (well, I speak for myself) remembered that, and that helped us get through things.
The big advantage of this option? It assumes good intent. You're not going into it thinking that the other person is wrong and you're right - that's never a good path to take. I only don't assume good intent if I'm 100% sure that the person is being purposely malicious. I give second - even third, or fourth - chances.
I didn't go into this discussion with a specific plan; the question of whether that was good or bad in that specific situation will remain unanswered. However, in general, I'd recommend having a plan for a one-on-one discussion. It's very easy to let your emotions get ahold of you and to then make rash decisions because of that. If you think this might be an issue, you can make a short list of talking points:
- What is the real issue at heart here? Is there an underlying issue beneath the continued disagreements?
- How did the problem come about? Don't try to assign blame; I have yet to see that lead to anything productive. Just the facts, ma'am, so to speak.
- What have the effects of the conflict been, and how seriously have they impacted the behavior of both parties on the site(s)? Have other users been affected?
- How can both of you work together to change this in the future? Are there still options left?
Unstructured discussions can have some merit. I got lucky before. However, its usually better to have something in your back pocket you can go back to if you want a successful, objective conversation.
Option 2: Disengage.
Let me be clear. The first option is always preferable, because at the very least, both parties know that the other wants a successful resolution to the conflict. Asking someone to work together to solve a dispute should ensure that you both respect each other - one of the most important results going forward.
Last year, I and several other users had a problem with someone on Physics Stack Exchange. He had repeatedly broken various rules and continued to complain and whine in chat, toeing the line and sometimes breaking it. I was occasionally a victim of his remarks, and I quickly learned that the best way to deal with him was to ignore him.
There are many possible venues for conflict in Stack Exchange. As I wrote before, we're all about conflict, but there are secondary means of communication outside normal questions and answers. These are, in my experience, where issues usually escalate.
Ignoring the person won't end the harassment. You may need to step back and let a moderator or other community member step in, especially if this user is in conflict with other users, showing signs of a large-scale problem.
- Chat. Just ignore them. Well, Ignore, with a capital "I", because there's a feature for that. Doing this to the fellow above worked. I let other mods deal with things when he broke chat rules.
- Harassment in comments. Flag away. Again, my case involved the same person as the chat harasser, attacking people in comments under an election nomination. I flagged, stepped back, and let the mods (in this case, also a CM, because this person was waaay out of control) take care of it.
- Meta issues. This is a place you won't be able to ignore the person as well. Meta is way more public. My suggestion: If they're being abusive, flag. If they're simply being overly contradictory . . . step back. Either their claims have merit or not, and if they're disagreeing just to disagree, the community will recognize this.
Do not take these last two actions unless the person is actually being abusive or harassing you! Make sure that you're not simply flagging because you disagree with them; your requests for intervention will likely be denied. Some might argue that that's flag abuse.