Most right-handed people are right-eye dominant and most left-handed people are left-eye dominant. But this certainly isn't true for everyone. For some people, hand and eye dominance are opposite - which creates a dilemma for participating in the shooting sports.
If you hold your hand out at arm's length and make a circle, then view an object across the room by looking through that circle, your brain must choose which eye will actually focus on the object. Since your eyes are roughly 3" (7,5cm) apart, both eyes cannot maintain the direct line-of-sight to the object. So one eye must take over, and you'll naturally position your hand more to the right if you're right-eye dominant, or slightly to the left if you're left-eye-dominant. In either case, the eye that takes over and maintains the sight-line is the dominant eye.
There is another easy test for this. While looking through your circle, close one of your eyes. If your hand appears to "shift" suddenly to the side, the eye you have closed is your dominant eye. If you close one eye and nothing really seems to happen, then you have closed your non-dominant (recessive) eye. Of course, you can see the object with either eye, providing you just use one eye at a time. But if both eyes are open, your dominant eye will always take-over and demand the sight-line through the circle.
So what does all this have to do with shooting a bow? In some cases, nothing. If you plan to shoot your bow by closing one eye and sighting with the other, it doesn't really matter whether your hand and eye dominance match or not. However, most serious archers sight and shoot their bows with BOTH EYES OPEN. Shooting with both eyes open gives you a much brighter and more natural field-of-view, even when looking through a peep sight. The dominant eye focuses crisply and does the actual "sighting" through the restricted area of the peep sight, and the other eye "fills in the gaps" in the picture (though not in complete binocular focus).
Field-of-View: One Eye Open
Field-of-View: Both Eyes Open
If you learn to shoot with both eyes open, you'll also notice another benefit. It's very easy to follow the flight of your arrow all the way to the target when shooting with both eyes open. But when you shoot with one eye closed, you'll often "lose" the arrow in flight because the peep sight jerks upwards at the shot, temporarily obscuring your line-of-sight. So shooting with both eyes open certainly has it's perks.
But there is a catch. If you plan to shoot your bow with BOTH EYES OPEN, you must choose a bow that matches your eye-dominance, rather than your hand-dominance. Again, this is because the dominant eye will always command the sight line when both eyes are open. We'll explain.
When you sight a bow, you basically have to manipulate your view to get all sighting elements in a straight line: beginning with your eye, then the peep, then the sight pin, then the target. So your dominant eye needs to be directly behind the string to line everything up. When the bow's sights are adjusted properly, your visual sight line should mimic the flight path of the arrow (not considering the arrow's parabolic arc). But with regards to the left and right aiming of the bow, the line-of-sight and path of the arrow should be the same.
When the bow is sighted with the dominant eye, everything works as it should. The line-of-sight and the path of the arrow are similar, and the arrow hits the target.
But if a shooter tries to sight the bow with the recessive eye while keeping both eyes open, the dominant eye takes over (even though it's not looking through the peep sight). The dominant eye commands the sight line automatically. This puts the first point of your line-of-sight, your eye, roughly 3 inches off to one side. To compensate, the shooter turns the bow dramatically towards the dominant eye. Unfortunately, when this happens, the sight-line and the arrow's path are no longer the same. The arrow misses dramatically left or right and "sighting-in" becomes all but impossible.
Truth is, right-hand bows are meant to be sighted with the right eye. Left-hand bows are meant to be sighted with the left-eye. And trying to force the issue otherwise is often a frustrating and futile act. Over the years, we've even witnessed a few shooters with eye-dominance conflicts try to lean over far enough to see through the peep sight with the outside eye. Of course, this creates an entire host of new ergonomic and shooting form problems, and it's a solution we strongly discourage.
For those with eye-dominance conflicts, the issue of right vs. left ultimately boils down to choice. If you are right-handed, you're of course a little stronger and more coordinated with your right arm. So naturally, you would want to shoot a right-handed bow. But, if you happen to be also be left-eye dominant, you have to make a choice. You must either shoot a right-handed bow - and learn to sight the bow with one-eye squinted closed, OR you must adapt to shoot a left-handed bow so you can keep both eyes open and sight with your dominant eye. It's a difficult choice, and there is no right or wrong answer. However, we do make the following general recommendations.
1. If you are a new shooter, by all means, buy a bow that matches your eye-dominance so you can learn to shoot with both eyes open. If you have no old habits to unlearn, you'll probably adapt quickly to the off-handed bow and never know it was supposed to be awkward.
2. But if you are a long-time shooter (archery, rifle, pistol, etc.), and you're already accustom to closing your dominant eye when you shoot, you may find changing to an off-handed bow to be particularly awkward and frustrating. In this case we recommend you not make the change and stick with your current technique (aka...dance with the one that brung 'ya).
Again, there isn't a right or wrong decision here. Success in the shooting sports is often a matter of subtleties in form and technique, and everyone is different. Whichever method you feel will benefit you most, and that will ultimately allow you to shoot most comfortably and accurately....that's the right choice for you.