I think I'd only want to know the specifics of the conversation if I had a need to know — if it was likely that not knowing was going to come back and cause trouble for me in some way.
The general question — if I'd want to know that SOMEBODY thought these bad things about me — is a little tougher. I'm a public enough person at this point in my life that I get to hear folks' negative opinions about me pretty frequently, one way or another, and what's most surprising about that experience to me is how little use I get out of most of those opinions.
If I'm doing something wrong, I want to know it. If I'm hurting someone maliciously or casually, I want to know it. If I'm offending people I don't mean to offend or expressing myself in a counter-productive way, my gut sense is that I want to know it. But nine times out of ten when I read or see some negative comment someone's made about me — even stuff that rises above the level of generic abuse, stuff that's got some specific content to it — it's of no use to me. Sometimes it's because I think the criticism is off the mark, sometimes it's just a matter of not being able to be everything to all people, sometimes it's stuff I'm already aware of and working on but haven't yet managed to fix.
And that stuff can clog me up. It can dull my instincts. A couple of days ago I happened to stumble across a months-old comment thread on a blog I didn't read on which a bunch of people I don't know were saying very positive things about my writing — specifically about pieces I'd gotten a lot of crap for. The folks in that thread had taken away useful stuff from what I'd written, and that utility for me far outweighed the crap. But for whatever reason, I'm usually more likely to see the crap, and too much of that stuff can tighten up my muscles and make me not write or not write with the kind of energy I want to be putting out.
So I guess the short answer is that if Person B thought that Person A was getting me wrong in a way that was widespread and fixable, I'd want to know that. If not, nah. I'm set.