cross dominant
Being cross-dominant is not a big problem in shooting, but it is hard for anyone who is NOT cross-dominant to understand, since there is no practical way to duplicate the phenomenon.
I am right handed and have a left master eye. I shoot handguns with the right hand and long guns off the right shoulder. With handguns, I use the left eye to sight with. (Out to 10 or 12 yards I have trained myself to shoot with both eyes open -- farther out than that and I have to close the weak eye.)
I use a Chapman stance (modified Weaver) that better brings the front sight of the handgun into alignment with my left master eye. I slightly shift my head to line my left eye up with the sights.
When shooting long guns, I close the left eye. This does cut down on peripheral vision to some extent, but I find it works a lot better for me than trying to shoot off the left shoulder, although I do practice bilateral shooting out to about 25 yards with both handguns and long guns because one never knows the precise nature of any situation one might find themselves in.
I do know some shooters with a cross dominant master eye who have learned to switch hands successfully.
I've taught police recruits at the local regional police academy off and on since 1988 and I have observecd that it was fairly common to have a left handed shooter who had a right master eye who also shot right handed, even though they used the left hand to write with. I suspect that circumstance forces left handed people to develop a greater degree of ambidextrous skills, simply because almost all firearms and many other implements are designed for use by right handed operators.
With my particular vision circumstance, I cannot use occluded eye gunsights with both eyes open.
I have found a number of people who cannot close a single eye independently. This condition pretty much requires that all weapons be fired using the hand/shoulder that corresponds to their dominant eye, regardless of which hand has greater dexterity.
Being cross dominant is not a big problem in most circumstances, but everybody's vision and level of physical skill varies. Experimentation will usually find the best option to select.