There are a few good points on this thread... But I have to make a few points here.
1) shooting with both eyes open is better that with one eye closed. Why? Because your face is more relaxed and your pupil is larger, allowing more light into the eye. This will NOT improve depth perception while looking through the peep sight. Everyone has what is called a "phoric posture" which is where the eyes go to rest when they are not seeing the same image (like when one eye is closed or in this case, one is looking through the peep and the other is not.) This posture is NOT pointed at the target, thus you will see double unless your eye suppresses one image... either way, you are not getting 3d or depth perception.
2) You will never be looking at the target with one eye and the pin with another. It is physically impossible. Also, you can not techincally ever see the target and the pin clearly at the same time... again, impossible.
3) Shooting with both eyes open is NOT inherently more accurate. line up the peep, the pin, and the target.... all else held equal, a 1 eye or 2 eye shot will hit the exact same point.
so WHY is it better?
As mentioned above, shooting with both eyes open allows you to relax at full draw. It also allows more light into the eye, again making for a more relaxed and possibly sharper target image. Keeping both eyes open during draw or while waiting at full draw will allow better field of view for movement or other things inthe peripheral vision. BUT, once you are behind the peep and aiming, your brain suppresses the image from your off eye and these seeming advantages are lost.
ALMOST ALL Pro archers shoot with both eyes open. BUT, if you look, many at the top shoot with blinders on their cap, tape on their glasses, or some other way of blocking the image to the off eye. Therefore, they are not getting better peripheral vision or depth perception. It does help, but not for many of the reasons mentioned. The comparison to shotgun or pistol shooting is not accurate, in these disciplines you are able to see the front sight (pistol) or target (shotgun) with BOTH eyes. With a rifle, it is similar, but much easier because one image is clear and magnified.
Why am I making a big deal out of this? I see it so often that people will stress over trying to shoot with both eyes open. I have even seen people try to shoot the opposite hand in order to accomplish this. I just think that there is more hype than is worth it. I just want to make sure that no one loses too much sleep.
LONG STORY SHORT: SHOOTING WITH BOTH EYES OPEN CAN IMPROVE YOUR FORM AND EXECUTION, THUS MAKING A MORE ACCURATE SHOT. THIS IS NOT TO SAY IT IS INHERENTLY MORE ACCURATE AND THAT YOU CANNOT ACHIEVE THE SAME AMOUNT OF ACCURACY IF YOU HAVE TO CLOSE ONE EYE.
Joe