I'm cross-dominant -- right-handed and left-eyed -- and it's no big deal.
When you shoot precise aimed shots using your weapon's sights, and you squint/close one eye (the one on your support side), your eye dominance ceases to matter, because you're no longer using binocular vision. You can also do the same thing when you're performing more dynamic defensive shooting or when shooting on the move, although you compromise your peripheral vision to some extent.
However, for optimal dynamic defensive shooting with a pistol, you typically shoot with both eyes open, and you simple adjust your gun position subtly for "both eyes open" sighted shots at full extension to fall in line with your dominant (support-side) eye.
Dynamic defensive shooting with a rifle is a bit more demanding for cross-dominant shooters, but with practice you can learn to pick up and pay attention to the images from your non-dominant shooting-side eye that your brain normally de-emphasizes -- all while maximizing your situational awareness by maintaining a wide field of binocular vision (i.e., shooting with both eyes open). Just spend part of your range time alternating between one-eyed and two-eyed rifle shooting at close ranges to develop and enhance this skill.
I'm Right Handed, Left eye Dominant. I shoot rifles righty with left eye closed because I can not focus on the sights, my focal points put the rear sight in my left eye and front sight in my left eye. No way around that except close one eye.
Pistol, I shoot right and sight left. It means I have to cock my head right FAR and although I keep the right eye open, my left must be centered behing the gun. It took about 3 months of training to learn it, and I still work on fast target acquisition so that I don't accidentally sight left. The key is to keep my head turned so my right eye doesn't try to jump in and overpower the left.
Archery, I learned left so that I could sight with my left eye (I should mention pistol and rifle per about I was right handed and learned how to shoot right, so I found it damn difficult to re-learn the skills left handed, easier to learn to switch eyes) With Archery, I started learning left and it took me three lessons over 4 weeks to finally get my goofy non-dominant left hand to learn and remember what it was supposed to do. Then, one day CLICK it worked. Now, I am as proficiant with bow-handling and nocking arrows lefty, you wouldn't know I was right handed.
PRACTICE what you can improve and work around what you naturally can't change. It'll come to you after regular, repeated training.
EDIT - People are suggesting closing one eye. This is ok if you have no other choice, but the advantage in maintaining a wider sight-picture and keeping your "other eye" (the one you want to close) open is a huge advantage tactically. I suggest practicing this and only this for a month. If you don't improve, then relegate yourself to squinting/closing.
For close range pistol shooting, in that case I'd focus on 'point shooting', which is all about body posture, reference movements, and feel. However that takes a lot of ammo and a flexible range to work on. I've not had the chance to do that myself, as the ranges I shoot at are very restictive. No 'holster work' allowed. Luckily for me my hands and eyes align.
However, something I can do is mentally choose which eye is dominant simply by thinking about it. Not sure how I developed that. I have at times wandered around for half an hour with one eye closed to see how depth perception does its job. Or those middle of the night bathroom trips, keep one eye closed in the light, then when I step out in the dark I have a good eye to walk back to bed in the dark.
I am cross dominate right handed and left eyed. Rifles with peep sites will tear you up as you get older. AK with open sites is best for us old timers. Pistols don't give me a problem.
If you want to confirm your eye dominance take a piece of paper 8 by 11 or so. Make a hole in the middle of the page about half the size of a dime. take the paper in both hands and hold it arms length in front of you and focus with both eyes open on an object like a spot on the wall or a picture. Close you right eye and see if you can still see the spot. Then open and close the other eye. you only focus with one eye. that will tell you quickly which eye you use.
I shoot with both eyes open, focus on the far sight, line up the sites, then focus back on the far site with the target in the background without focusing on the target.
Shooting with both eyes open IMHO and my training has taught me to utilize my peripheral vision more, allows you to recover quicker on target from recoil, and allows you to see possible attackers.