When hanging tandem, I use 26" spreader bars on the hammocks. It's enough to keep both rigs under one tarp, but occassional contact happens. 36" might be better for a permanent setup. You could each take turns in a hammock and measure the max space (width) you occupy while lying in various positions. The spacing would be the largest measurement + a few inches. If wall space allows, instead of hanging exactly parallel, you could position the head ends a little closer together to allow for good morning kisses form the hammock
A particular anchor may spec out at needing just one or two anchors, but what will give you peace of mind each night in the long term? None of these fasteners/plates are very expensive. If you've got a proper hammer drill (bits with SDS Plus shanks) setting anchors is easy.
Another option instead of a plate and ring is to mount a piece of slotted angle to the wall vertically. This will allow you to adjust the hang angle/hammock height more easily. Attached is a pic of what I did in my daughter's room.