I have been a firearms instructor since 1969, first in law enforcement, then for the past 16 years, doing civilian concealed weapons classes, too. For about the last 6 years, only civilians.
With a handgun, there really seems to be no difference in how well someone shoots when he or she is right handed and left eye dominant. Probably that is because from a Weaver-type stance or isosceles stance, there is so little movement of the gun necessary to align the left eye with the sights. A bit more with the Weaver, really pretty small with the isosceles.
I have, in the distant past, seem some pretty good bullseye shooters who were right handed, left eye dominant.
That changes dramatically with long guns, especially shotguns.
I have forgotten the percentage, but I believe (too lazy to look it up) that in trap, more than half of the women shooters are left eye dominant (I kind of recall it is more like 75%).
On of our greatest trap shooters, Phil Kiner, has done a lot of work on cross-dominance. He was very helpful in giving me advice for my wife's cross-dominance problem.
And, there have been several different attempts at solutions.
1. Bending the stock so the left eye aligns with the beads. Usually unsuccessful, and costly to bend a heavy shotgun stock enough for that to happen. Looks weird, too.
2. Shoot left handed. Works for some, but not most.
3. Shoot right handed and cover the left eye shooting lens with:
A. An opaque dot, or scotch tape-works for many.
B. A patch of some sort-again works for many-and my wife shoots trap like that. But, she is careful to conceal the patch under her cap and only drops it over her eye when she is on the line and ready to shoot. Otherwise, she thinks she looks like a pirate.
C. Just close the left eye. Seems to work for most. Some shooters just can't do that while shooting at clay targets.
A & B allow the one eyed shooter to still keep both eyes open, comfortable and still give some depth perception.
There are a lot of "one-eyed shooters" who are great shooters.
My first instructor, Nora Ross, is an All-American, one-eyed shooter. Amazing lady.
The problem with a shotgun is you are looking at the target, not the sights and, of course, the target is moving at over 40 miles per hour.
Keep both eyes open and the dominant eye takes over. Since the eye is the "rear sight," if you are shooting right handed and the left eye is the rear sight, you simply miss. And, that happens even though you don't actually use the "sight"
to shoot.
I am a two-eyed shooter, being right eye dominant, so I am blessed not to have these problems.
Bob