Left handed right eye dominant here. It really depends on what I'm shooting as to how I'm going to shoot as far as closing one eye. I found that shooting from the right shoulder using the dominant eye is pretty easy, especially since that's just how I learned to shoot.
When using my pistols I will typically close one eye, unless I'm training for more practical circumstances and not working on accuracy, then I'll keep both open at times. When using high power scopes, definitely close one eye. When using something with less power (typically 5x or less) I can shoot just fine with both eyes open. Shotguns are a toss up. I don't really pay attention to closing one eye or not and will shoot one day with both eyes and close one eye the next day without even thinking about it and have the same results as far as accuracy goes.