I shoot right handed. Right eye dominant (though that eye isn't as good as my other one :-? ). Have not tried shooting left handed. It feels uncomfortable and out of place. Even with long guns, I suck left-handed/it doesn't feel right. In the future, I'll work on this.
My barber's father had an accident with a bungee strap last spring and lost almost all sight in his dominant right eye. Now, he has to get used to shooting bow's and firearms left handed or just left-eyed and had been having a difficult time (he's in his mid-70's). With some new sights, he was able to get used to it and was able to get a couple of deer this year.