Human eye dominance comes in the same spectrum as human hand dominance.
For hand dominance, people can be
1) strongly left- or right-handed,
2) mostly left- or right-handed,
3) mixed dominance (doing some tasks with the right hand, others with the left)
4) truly ambidextrous (rare! -- doing all tasks equally well with either hand)
It's pretty much the same for eye dominance. People can be
1) strongly left- or right- eyed
2) mostly left- or right-eyed
3) mixed dominance (sometimes one eye takes precedence, sometimes the other; which eye takes precedence will vary with conditions)
4) truly ambiocular (can keep both eyes open while consciously choosing which eye takes precedence for any given task)
For people who are in category 1), it is easy to learn to shoot with both eyes open. Might take a little bit of practice, but mostly all it takes for these people is hearing that it is possible. Even for these people, however, eye dominance can sometimes switch depending on the lighting conditions and angles. When that happens, they will have a shot that goes off to one side of the target.
For people in category 2), it will take a lot more work. And frankly, the amount of work it takes may not be worth it, especially since when the eyes are more evenly balanced it's even more likely that the 'non dominant' eye will suddenly take over.
For people in category 3), keeping both eyes open is asking for trouble. Because these people do not reliably know which eye will take dominance at any given time, their eye dominance jumps on a regular basis. They can't even practice both eyes open at the range on a calm day, let alone learn to trust it in chaotic, rapidly changing self defense conditions. Again, when eye dominance shifts, the shooter will often erratically and unexpectedly shoot several inches or more to one side even when their trigger control is excellent and their grip on the gun solid.
For people in category 4), these folks are the lucky ones. They can switch eyes when they switch hands (everyone who practices for self defense should be able to shoot with either hand). They can easily use the right eye around right side cover and the left eye around left side cover, thus keeping more of themselves behind cover. And like the people in category 1), they may not understand what all the fuss is about.
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For most people, there's no real benefit to keeping both eyes open while shooting modern handguns for self defense.
Yes, tunnel vision is a thing that happens. But tunnel vision is not a vision issue. It is an "attentional focus" issue -- in other words, it's a brain issue rather than a eyeball one.
While it's very, very important to open your eyes and look around as soon as you're done shooting, you cannot simultaneously pay strict attention to getting your hits and see what's going on to the sides of you at the same time. Brains are just not wired that way. That kind of multi-tasking is not within human capability (which is why people look down at their phones, or turn off the radio in order to concentrate -- we 'tunnel in' all the freaking time. It's not some rare or weird thing. It's normal processing, just on steroids.)
Anyway... The answer is NOT to multi task, but to thin slice. Do the thing that needs to be done RIGHT NOW, do that thing with 100% of your focus, and then move to the next thing as fast as you capably can.
What's that one thing that's most important RIGHT NOW? Stopping that guy from killing you. Do that. Use your eyes however you need to use them in order to see so you can do that. Then look around and see if the bad guy has friends. Don't dawdle.
Categories 3) and 4) are both rare, about half as common as mixed hand dominance and true ambidexterity in the general population.
With modern, self-defense handguns, there's no disadvantage to being "cross dominant" -- that is, to shooting with your right hand while being left-eye dominant.
It makes no difference for shooting defensive handguns. It used to matter a great deal, back in the days when people shot bullseye by sighting down the arm. And it matters for long guns. But that's not what we're talking about here.
pax