The way my fiance and I hung our hammocks last time we went out was to put the attachment points on either side of the tree. Facing the "head" tree, my Dutch clip was on the right side of the tree and hers was on the left side. This made it so our hammocks were hanging about a foot apart. The bigger the tree (diameter) the farther apart the hammocks would be. Obviously this wouldn't work too well on a smaller diameter tree.
Neat idea! Let us know how the testing goes... I'm def interested as me and my better half like to take trips together as much as we can.