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There can be more going on here than just "dominance" of one eye over the other. When I first started shooting handguns, I always shot with one eye closed because I couldn't use the sights properly otherwise. When taking some serious self defense/firearms training (40+ hours and a couple thousand rounds over two weekends, including night shooting well into the evenings) instructors tried to get me to shoot with both eyes open. Focusing on the front sight like everybody teaches, I see two targets, of course, and two rear sights, superimposed so as to obscure the notch in the rear sight. No way could I get what most people apparently take for granted as a normal sight picture. Instructors had me do some tests and concluded neither eye appeared to be particularly dominant. Saw an eye doctor later, who is also a competitive shooter. She did her own tests and concluded that between dominance issues and strong astigmatism, I would probably never be able to use iron sights normally with both eyes open. Her recommendation to me was this: Forget both eyes open. You can't "train" your eyes or brain to correct this any more than you can train yourself to be 6'2" instead of 5'9". Your eyes don't work the way many other people's eyes do. Close one eye to use the iron sights or get a laser or other optical sight for the handgun.
I opted for laser sights and have CT Laser Grips mounted on 3 of my own SIG's and my wife's Springfield Armory XD-40. Couldn't be happier. Put the dot on the target and shoot. Keep both eyes open and focus on the target, not the sights. Way better than trying to line up iron sights under the stress of a gunfight. Dramatically improves ability to hit the target. Police and SWAT teams are increasingly using laser sights for exactly this reason, and their eyes probably work just fine. And I'm seeing lots of photos of Aimpoints and other optics mounted on military rifles deployed in Iraq and other dangerous places.
I do include laser-off practice, with one eye closed, but decided not to sweat or agonize over the issue. Use the laser and go from there.