Two eyes are better than one for me, for the following reasons.
I don't get a headache. I see much better, pick up the targets sooner (especially left going targets), and because I don't have to hold as low, there is much less gun movement to the target.
If I dot or tape my left lens, my left eye gets sore trying to focus. It also tries to see around the occlusion. When I wear an eye patch, the "picture" my brain sees is darker because the dark gray image of the eye patch my left eye sees overlays the image from my right eye.
I can't close my left eye without squinting my right. I can't stand walking with my left lens taped, or while wearing an eye patch- even from post to post. I don't like shooting with one eye. Actually, I hate shooting with one eye. Actually, I'd rather quit shooting than shoot with one eye.
Phil, I don't have a dominant eye. It has always been a problem. I spent a lot of years shooting one eyed until I finally decided i was going to learn to shoot two eyed no matter what. It was a frustrating effort. I eventually learned to do it, and will never go back. However, there were many problems to overcome even after I had "learned". Gun fit became even more important. My left eye occasionally locked on the bird and I'd miss inexplicably. Phil Kiner picked that up and made me aware of it. I tried all of his suggested fixes and stuck with the ones that worked for me.
I've worked my way around most of the problems, but I know I will never be an AA shooter. I take it as a fact of life than an average of 4 times an event, my left eye will take over and I'll miss way left. That's why I'm a high B, low A class shooter, and that's where I'll probably stay. I'm okay with that.
For disciplines other than trap, two eyed is much, much better. I cannot imagine hunting pheasants with one eye. Nor can I imagine completing a sporting course using one eye.