You close one eye when you're firing a rifle. You keep both eyes open if you're firing a handgun.
So Sayeth the guy that was in Iraq and in firefights.
The eye closed for rifle is that you know the size of a human, and thus can determine distance by the apparent size of a person (we don't kill no little people). No depth perception required. For hand guns, you align your sites with your dominant eye, but you need your second eye open because a hand gun you can tilt all sorts of funny ways, whereas a rifle you tend to socket into your shoulder when firing. The second eye open allows you to determine the trajectory of the weapon, because even the slightest change in grip and posture can change your site picture, thus you have to "eyeball" firing with a handgun.
Case in point, I am awesome with a rifle. Caroline sucks at it because she can't close one eye. I am the suck at handguns. Caroline can is freaky/scary good with handguns, because you can fire with both eyes open.
Seriously... she's fucking scary with a handgun.