On my server, running champions in Queensdale provides much enjoyment for a large number of people. New and experienced players alike band together running between the various champions.
Yes, there are occasional squabbles when someone starts a champion out of rotation. This is usually a simple misunderstanding: for example, a newbie sees the troll stone and picks it up, and others who were standing around fight the troll since it’s trying to kill them. Other times cross-communication in map chat causes two things to be engaged at once; these issues get sorted out quickly enough, since normal people want to maximize their rewards if they’re going to be spending any real time in the zone, and join the larger group.
When I see real arguments in map chat, it’s always between a large group of players who are enjoying the camaraderie of working together, and a single person who is aware of the group, starts several champion fights out of the “rotation”, and then makes it clear through his responses in chat when others express dismay that he’s a sociopath and doing it because he enjoys being a jerk in a situation where repercussions are impossible.
If you’re level 80, you’re a tourist in the zone like everyone else, and unless you just happen to smack down a single champion because you run past it on your way to some other task (e.g., you kill the spider on the way to the Demongrub Pits, or the Oak when you’ve teleported in for the Shadow Behemoth), there’s no reason to repeatedly frustrate the others who are chain-killing the champions other than pure spite, because you’re gimping your own rewards (along with everyone else’s) by doing this.
No, nobody owns the champions. But I don’t think there’s any real problem to be solved, at least on my server.