Yes, we do ours out of eye-shot of neighbors as well. And (here you may think we're crazy) we also do it out-of-view of our flock. They get put in the run when we do the deed. They get let out when we're done with the plucking and removal of the feet and head.
We hang ours on a wall of the shed. Plastic garbage bags stapled up help protect the wall. An occasional spurt does reach unprotected walls, but in our case it doesn't really matter; you may feel differently.