Right Sean, not a political issue at all. Neighbors can choose to cooperate whether they are fascists or Luddites. It just makes sense where space allows to use vegetation to define borders. Consider that in the city we are affected by borrowed landscapes more than the ones we own.
I've seen shared hedges and trees bring people together if they bring a positive attitude into it. But yes if their thinking is stinking then any trifle is a battleground. "Private property rights" trigger some weird aggression over here too.
Liz, it sounds like the vine is a bigger problem than the tree right now. By using strings and trellises, vines can be useful screen with no cost other than a snip now and then.
Can you post a picture?