I was taught very early in life to keep both eyes open when shooting anything---rifle or shotgun. I am able to concentrate enough to see only with my right eye if I need to(I'm right handed) just to line things up or look through a riflescope, but both eyes are always open.
As I said, I was taught this when I was Just a little fella, and it has worked well for me. It certainly helps when you are bird hunting, or shooting clays, and I used to shoot a LOT of trap.
I have known people who for one reason or the other, have had to switch from right-hand, right eye shooting to left-hand, left eye shooting, and most wore either an eyepatch or as was stated before, some put a piece of tape over the lens of their shooting glasses to change the dominant eye effect. It's whatever works best for them.