I'm the opposite: left handed and right eye dominant. I use my left side for pistols and my right for rifles (also archery, golf, and two handed swords).
I've tried shooting precision rifle left handed, but I always have issues getting a good sight picture. I'd recommend spending a lot of time behind a rifle just looking through the scope. Move positions and look through the scope again. Then switch to the other side and try again. You might need to adjust the position of the optic to get the best eye relief for that eye.
If you find that one side has a much easier time getting a good sight picture (like I do) then that's the side to shoot with primarily. If both sides are great, then I'd make the choice based on which side feels better ergonomically (holding a rifle in my left hand and changing mags with my right feels weird, even though that's what I do with a pistol). I wouldn't choose right handed simply to have a better selection of rifles, as some have suggested, because there are enough ambidextrous or left handed options, especially in the higher end market.