HammsBeer is right on.
With handguns there is no discernable problem with shooting cross-dominantly. You achieve a strong-hand grip but you'll "yaw" the gun ever so slightly, to line up the sights with your dominant eye. Many/most shooters will do this without even realizing it.
If you are able, teach yourself to shoot from the strong-eye shoulder with long guns. It turns out that most folks can train themselves to be perfectly proficient operating long-guns with their "weak" hand on the trigger, more easily than they can force their non-dominant eye to take over. Wing-shooting with a shotgun, cross-dominantly will be an exercise in futility. You CAN make do by closing your dominant eye, or occluding it with some clear tape on your shooting glasses.
But think about that for a moment. You just took your GOOD eye, the better half of your vision, out of commission. Shooting is all about SEEING. How much better to use your best eye -- and even better yet, to use BOTH? (Not to mention the huge loss of balance and depth-perception that closing an eye brings.)
Some folks just have so much time behind the gun "wrong-shouldered" that they don't feel that they can learn to do things any other way, and that's understandable. But if you're able to make the switch you'll be a better, more natural shot, especially on any moving targets and/or dynamic situations.