I shoot with only my right eye open because I am left eye dominant and I shoot a right-handed bow. If I did not then I would have difficulty looking through the peep sight as my left eye would constantly fight to take over the focus. However, for those that do shoot with their dominant eye then having two eyes open is reportedly the way to go. The reasons? I honestly do not remember them as I have not had the motivation to explore the issue further.

Sorry I could not help more.