I’ve sometimes seen it said that in order to have a productive discussion, people on both sides need to be willing to change their minds. I think that’s probably slightly overdetermined– you can find examples of cases in which neither side was going to change, but they managed to sustain a mutually beneficial dialogue all the same. The physics example that comes to mind is Bohr and Einstein, who spent decades arguing with each other over the philosophical basis of quantum theory, but were nonetheless good friends. They pushed each other, forcing each of them to refine their arguments and think deeply about the nature of the universe, and in the process broadened our understanding of reality. Neither really budged, but everybody benefitted from the exchange.
I think the real minimum condition is a belief that both sides of the discussion are being carried on by reasonable people arguing in good faith. That is, the people on both sides are sincere in their statements, know their own minds, and are doing their best to behave in an ethical manner. They’re not taking extreme positions just to provoke people, they’re not cynically saying things that they don’t believe but think will sound good, and they’re not working toward morally repugnant goals (the enslavement or extermination of large groups of people, for example). People on both sides need to accept that their opponents are intelligent people who hold their beliefs for reasons that they find valid.
It’s a little difficult to see where to draw the line on this, but I think you’re pretty clearly across it by the time you start exchanging accusations of bigotry.
If you honestly believe that your opponent is a bigot, you’re excluding the possibility that they could be a reasonable person arguing in good faith. They are sufficiently motivated by prejudice that you cannot be working toward mutually acceptable goals, and their arguments cannot be trusted to be sincere and valid.
If you don’t believe that your opponent is a bigot, but say so anyway in order to score rhetorical points, then you’re the one arguing in bad faith. That’s pretty much the definition of arguing in bad faith, really– making accusations about your opponent’s character that you don’t really believe in a cynical attempt to provoke them or to sway third parties.
In either case, you’ve closed off any possibility of a mutually beneficial exchange. There may be other reasons to carry on such a conversation– for the entertainment or edification of third parties, or from a childish love of provoking other people– but some time before you get to calling each other bigots, you’ve moved out of the realm of productive discussion, and into some very different territory. And it’s not always clear that anyone will benefit from the exchange.
You can have a productive conversation in which no minds are changed but everybody benefits, but you can’t do it when you believe that your opponent is an imbecile, a lunatic, or a bigot. And if you think your opponent is an imbecile, a lunatic, or a bigot, you should think carefully about whether this is really a conversation worth having.