Ahoy matey. I figured I would try to beat everyone to the punch. Are you left eye dominate? If so, keeping both eyes open, as suggested, should wok. If not, do what ever is most comfortable. I wouldn't care about getting stares.
One thing snipers do is drape a cover over there heads. This serves two purposes. One it helps to conceal their heads and maintain stealth. Others use it as a form allowing their eyes to relax while scoping their position. Being in the position they have to be in for an extended period of time forces them to observe with both eyes open. If you can't master that practice then the head cover might be the option to help your eyes relax while shooting and prevent double vision.
Correct target shooting should be done with both eyes open, and the "non shooting" eye covered with a patch or shield...
This patch allows the user to flip it down when shooting and flip it up when not, for reloading magazines, scoring, etc.
Some shooters believe an eye patch should still allow light to enter the non shooting eye to keep the aiming eyes iris normal. By closing the non shooting eye the aiming eye will adjust for the lack of light entering and you will lose your focus, resulting in poor scores.
I have heard of shooters making a patch that hangs down in front of the none shooting eye, out of `milk jug carton` or the opaque plastic bottles we have here ( not sure what you guys use)