Have to disagree with you Brian. It is possible to "get by" with a scope sighted rifle by closing one eye, but you are working with a handicap. If you also shoot a shotgun you are working with a severe handicap trying to shoot with your dominate eye closed. You are cutting your field of view in half with one eye closed making it harder to pick up a target. With only 1 eye open you severely limit depth perception making it harder to judge the distance, speed and angle a target or bird is coming from.
These are less of a concern with a scope sighted rifle, but can be a factor in certain situations. To be succssful with a shotgun you really need to be shooting with both eyes open and with the gun mounted on the same side as your dominate eye. No reason to shoot a shotgun from one side and a rifle from the other.
We went through this with my brother, it took some time, but after he mastered shooting lefty his accuracy improved dramatically.