Hold both arms at length above your line of sight........make a small hole with both hands clasped together.
Now focus on an object in the distance......lower your hands (Still at arms length) and center that object in the hole you created with your hands. Close and open your right eye, and then your left eye. Whichever eye allows you to see the object in the hole is your dominant eye.

I'm left eye dominant and shoot right handed, it has never been a problem for me shooting long guns or bows, and I shoot a bow with both eyes open.
Where I really notice it is shooting a handgun with open sights........I naturally want to line the rear sight up with my left eye. I can shoot pretty well either way, but the left eye is definitely wrong.
I can shoot fine right eye with a handgun at the range because I am thinking about it, but I'm 100% sure that in a combat/defensive position where I had to shoot fast and/or instinctive that I would line up my left eye.