Cross dominant shooting is a funny thing, I'm left eye dominant and right handed and I've noticed the following:
With pistols, I shoot more accurately using my left eye, but I've always been center when shooting with my right eye only, I just shoot tighter groups closing my left eye.
With pistols, I can get good two eyes open sight picture by indexing my chin on my right shoulder and score good quick hits and maintain more peripheral vision than shooting with one eye open and one eye closed. Some like to cant the gun to the left at around a 45 degree angle (plus or minus) and have good results, but I didn't like the way the gun handled during recoil in that position.
With rifles I've always shot right eyed only and managed to qualify mid to high expert multiple times while in the Marines, that's shooting with iron sights out to 500 yards and scoring hits on man size targets, shooting at 300 yards and 200 yards. It's not that big a deal being cross dominant with rifles if you're using iron sights. Some issues start up when you being using 1X optics, I can NOT get a good sight picture (proper sight alignment applied to the target) with an Aimpoint or EOTECH unless I close my left eye. Again canting the firearm will probably help, but I haven't really tried it with an optic equipped rifle. So with rifles I will shoot right eye only.
As a cross dominant shooter, if you are going to shoot trap or skeet using your strong hand, you're screwed. You need to shoot the shotgun with whatever hand matches to your eye. You can probably learn to do it with the other hand, but it's going to be a pain in the rear.
Back to pistols as this is the HANDGUN forum...
There are many different schools of thought on cross dominance, I've been looking into this for a couple years and reading what others have to say on this matter and I've seen the following arguments.
1: You should shoot with both eyes open for the improved peripheral vision.
That only works if you can actually see your sights with both eyes open, unless your point shooting you need to see your sights to make hits. Not to mention that in times of mass adrenaline, we have the tenancy to go into tunnel vision and that improved vision of having two eyes open may not matter at all.
2: You should shoot with whatever hand corresponds to your dominant eye.
I don't know about everyone else, but my left hand is damn near useless. But here are two more points, one for and one against the above statement.
A: Using your off hand that corresponds to your dominant eye will leave your strong hand available for negotiating obstacles and your mag changes may be faster.
B: If you've been shooting with your hand that doesn't match your dominant eye, it's going to take a lot of practice getting used to drawing, aiming, and firing with the same precision* that you get with your dominant hand.
3: You should cant the gun closer to your dominant eye.
As noted above, guns are meant to recoil on a vertical plane, that's usually means that the gun is going to go with the front sight and travel at an arc. Canting the gun to get better sight picture is aligning the gun to hit you in the face and take your sights further off target while firing.
So here's the real deal from what I've tried and what I've learned from other instructors.
Use whatever method works for you and improve that method.
If you find that you get better hits and better performance using your right eye only, stick with it. If your left eye is dominant but you close it, your right eye is now dominant by lack of options.
If you want to shoot with two eyes open, adjust your head laterally until you find the right sight picture and practice alot with it until that position is where you naturally end up after drawing and presenting the gun on target. It's taken me almost two years but the practice is finally paying off and I'm getting that "instinctual muscle memory" sometimes I lapse and find myself closing my right eye but for the most part I'm doing OK.
If canting the gun works for you, use it.
You don't have to shoot the same for every ocassion. If I'm shooting a rimfire bulls eye competition where I'm not over working my left hand, I'll shoot left handed as that's where I get my most accuracy. If I'm shooting close up defensive stlye I use both eyes open, if the target is a little further out and smaller, I will close my right eye for the most accuracy while using my dominant hand as it gives my best overall performance under prolonged use.
There is no right or wrong way to shoot if the results you get with a given method work for you.
* Accuracy vs. Precision: Accuracy is doing somehting correctly, precision is doing something accurately a number of times in a row and acheiving the same result at the end of an action. I can shoot more accurately left handed, but as time goes on, my left arm wears out faster than my right and my accuracy is effected as a result of "the shakes". While shooting right handed, I acheive an acceptable level of accuracy and due to the greater strength of my right side, I get more precision, that is to say I can acheive the desired result (a good hit) more often while shooting right handed.
