I genuinely believe it's possible to shoot with whichever eye is dominant, regardless of hand dominance. Personally, I am neither eye dominant, and that seems to SERIOUSLY throw just about everyone, including county coaches.
My string sits on the tip of my nose, not either side, and I find that if I let either eye gain dominance, which can happen if I start to think about it, I really can't hit a barn door. With both eyes together, I'm pretty good.
I thought Aslan's thoughts on the subject were genuinely interesting, and definitely well worth the read.