I will start out by saying, "If it ain't broke, don't fix it." Don't mess with a winning formula one way or the other. However, for those who are just starting out and looking for some input on which method to use, here are some suggestions shooting your bow with one eye versus two.
There are few things more important during the shot than your sight picture. And there are few things that have more influence on your sight picture than your eyelids. Sounds crazy, but you have several options here: you can shoot with both eyes open, with your non-dominant eye fully closed or with your non-dominant eye partially closed (squinted - that is what I do). Here are a few tradeoffs to consider when deciding which style to choose.
BOTH EYES OPEN
There is no question that shooting with both eyes wide open produces the widest field of view, but there is a potential downside. If your aiming eye is not significantly more dominant than your other eye, your eyes will fight to determine which one controls the sight picture. The result: as situations change, your sight picture will change too.
Dominance becomes an even larger problem when the light is low and the restriction of your peep sight slightly diminishes the acuity of your aiming eye. At times like this, it is very common for the non-aiming eye to seize total control of the sight picture. When that happens you'll miss by a mile. You can learn a lot about visual acuity and dominance by practicing under low light conditions for a couple of days.
NON-DOMINANT EYE COMPLETELY CLOSED
This the way Randy Ulmer shoots. In fact, when he competes he uses a blinder for his left (non-dominant) eye. His goal is to eliminate all possible variables in the sight picture, and by simply closing his non-aiming eye while shooting, he removes it from the equation. Sure, he gives up some field of view, but he says that he is so focused on the pin and the target that he really doesn't want to be distracted by anything on the periphery of his sight picture anyway. He doesn't consider the lost field of view to be a negative.
NON-DOMINANT EYE PARTIALLY CLOSED
This is the way I shoot. I have found a good compromise by squinting my non-aiming eye. This permits a fuller field of view while greatly reducing the acuity and possible dominance of the non-aiming eye. A possible lack of consistency is my only concern with this style of aiming but I have done it for 20 years, so it just happens naturally now. I never even think about it.
It will work fine as long as you always squint the same. Ulmer states that at tournaments he has seen shooters whose aiming style changes as they get tired or when they're under pressure. When your sight picture changes, your accuracy has the potential to change too. So if you use the squint method, you have to be more diligent to be sure you are doing it the same on every shot.
There are a lot of ways to aim, and my choice is but one of them. In the final analysis, consistency is the key to all aspects of archery and aiming is no different. After you've experimented to find the best aiming style for you; keep it exactly the same on every shot. Be conscious of your eyelids; these simple shutters can have a major effect on your accuracy.