Thanks for the suggestions all! I think I will assume that I can hang the hammocks from either side of the tree and depending on the diameters of the trees, will consider staggering them.
I also had another thought about this.. I like the idea of hanging from either side of trees and am wondering about whether I could create some sort of "extender" that would increase the diameter of the tree, like a short piece of firewood.. the only concern would be if the extender would fall out from the webbing. You would somehow need to cinch the webbing down rather than the usual loop through a biner.
Also on a confessionary note, a good portion of my motivation for this is that I want to get another hammock for myself (she will only come with me on car camps), so it's all about dual purpose in gear