I'm cross-eye dominant (right handed, left-eyed).
When shooting pistols, I'd shoot right handed, but sight with my left eye. I pretty regularly keep both eyes open when shooting pistols. Is that something most have trouble with? I'd think in a defensive (or offensive) situation, being able to aim and keep both eyes open is pretty important.
With rifles, I shoot right handed, and I (obviously) use my weaker right eye. I've never had a problem doing this either. Again, both eyes open pretty much all the time, unless I'm taking some extra care in placing a scoped shot.
Now here's the interesting part for me: Last year I was in an auto accident where a blow to the head left me with a small blind spot in my left (strong) eye. The spot is pretty small, but it's right near the center of my vision. This forced me to change the way I shoot handguns. No more sighting with the left eye - I had to change to my right (weaker) eye. It wasn't too hard to get used to (probably because I had no choice). My shooting suffered slightly for a while, but it's gotten better now. The hardest part was the muscle memory that required changing. When quickly bringing the pistol up to ready, I would instinctively align the sights to my left eye.