I prefer to close one eye. That's how I've grown up shooting rifles and handguns, and that's what I like.
I've tried keeping both eyes open, and I find it confusing to view a scene with no magnification in one eye while seeing the same scene magnified through the scope with the other eye. My brain doesn't like that at all.
And my eyes are pretty much equally dominant, so because I haven't practiced scoping with both eyes open, it can take some time to tell my brain to concentrate more on what my right eye's seeing... therefore, it's much faster and easier for me just to close my left eye. And sometimes speed is everything when you're hunting.
For folks who are used to it and who like it, there's nothing wrong with keeping both eyes open. If it works for them, then that's great, more power to 'em. For some shooters, keeping both eyes open can help with target acquisition, and if that works for you, then by all means do it.
When using any scope for long periods, I sometimes have to relax my left eye and hold it closed with a finger, rather than squinting it shut with my facial muscles. Squinting can make me feel strained, but leaving the eye open befuddles my brain. This is usually more of an issue when using a spotting scope rather than a rifle scope.
In summary, whichever method works best for you is the one you should use. It's a matter of taste and opinion, and we all know that those vary widely among any group of human beings.
- Russ Chastain