Just my personal experience, I am left handed, right eye dominant.
I shoot both short and long weapons left handed.
Shooting SMGs/rifles ( and pistols ) both eyes open has not been an issue, atleast I have not seen it as something that would hinder my performance with long guns, either with iron sights or red dot / variable power scopes.
In the Army I shot well in the range quals and due to my civilian shooting background, our instructor was pushing me for a slot in the sniper NCO course, but I declined that offer to go to a another unit. Also shot well in our applied training shoots later on.
I shoot both pistol and rifle IPSC at a national level and have top 5 placements in both, and like said above, have not noticed any limitation to my performance from cross eye dominance.
The only issue that I ran into to recently was putting a small Burris red dot sight to the lower left side of my S&B Short Dot, there the combination of a large side turret and rear ocular of the scope messed with my eyes and I will put the dot at the top, to 11 o'clock, whilst still maintaining some cheekweld.