My current position is that Tatoeba should not get involved in personal conflicts. Unfortunately, we cannot prevent them from happening, but all we can do is to remind people that this is not a warzone, this is not high school, and everyone has to make an effort to behave as a civilized human being.
A few good practices to remember are:
1) To NEVER reply to provocation with more provocation, even if it's tempting.
2) To ALWAYS solve personal conflicts in private, not in public.
We also have to understand that whenever tensions appear, everyone who is part of the conflict has their own responsibility for it. It really makes it easier to solve conflicts when people can, for once, forget about their ego, stop looking at others flaws and instead, look at their own mistakes and admit them.
At any rate, I don't want to threaten people with censorship or punishment, because this is not the solution. Tensions originate first of all from misunderstandings, so the only real solution is to lift up these misunderstandings... which requires an effort of communication.
The best thing moderators can do is point out to rule #14 of the contributors' guide (which I will reformulate to include more thoughts on good behavior).
If despite of everything, someone wants to continue insulting other members, well... Technically they can. But socially, they can't. Because by doing so, they will generate a terrible image of themselves, they will lose everyone's respect and they will be completely ignored since no one will want to waste time talking to them. I don't think anyone in their right mind would want to end up like that...