IMO, sorting out your eye dominance is going to be the tough bit.
Having at least minimal skill with your weak side is a good idea.
That being said, treat shooting from each side as a completely separate thing. The biomechanical shooting machine that is you and your gun will work a little differently, therefore it will FEEL different, and the results will be adequate, but not exactly the same.
If you approach the excercise expecting the same biological feedback cues from each side, you'll wind up barking up the wrong tree.
By way of explanation, as we learn our shooting skills, we amass a checklist of physical sensation cues that tell us when we're doing things right. Arm position here, pressure in cheek there, shoulder here, visual picture thus, etc. When something's not checked off on the list, we automatically compensate until they are. Your "other" side shooting checklist will be DIFFERENT.