As I teach students in my pistol class, I come across some that are right handed but left eye dominant or the opposite. What I suspect is that the right handed people who are left eye dominant are really "closet lefties" meaning that they were taught to write with their right hands even though they were really left handed. Still, they flounder when asked to use their left hand when left-eye dominent. They really only have two choices that are practical. First, they can learn to shoot lefty using their dominant eye. Second they can learn to use their non-dominant eye and shoot righty. There's no harm in that except that they'll have to work really hard to get good scores if they shoot competively.
There is a third option that is not a good one to practice although I've seen it occur. That is to shoot cross-eye dominent. That is to use your left eye when holding the handgun in your right hand or the other way 'round.
There is a well known actor who play a cop named "Horatio" on CSI Miami. Watch that show sometime and pay attention to him when he pulls his fake gun. He's clearly siting that pistol with his left eye even though he's right handed. It can be done but it's not an efficient way to shoot because it's hard to shoot both hands when having learned that technique.
Ultimately we should all be almost as proficient with our weak side as with our strong side. You'll never know if you're ever called upon to use your weak side. Today most modern handguns are ambidextrous so it's not that hard to do. And, if you've never tried your weak side, you may be really surprised as to how well you actually do! Because you haven't practiced with that other hand, you have no muscle memory to hurt your targeting. It's a good chance to relearn how to shoot all over again.
Finally, you can only focus on your front site or your target. It's impossible to do both. With practice you'll learn that. Forget about the rear site and just put the 'pumpkin on the fencepost" with your dominant eye and the same hand and you'll be successful. Poor practice won't help, however, so it might be helpful to have a good instructor watch you as you shoot and help correct your deficiences.
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