I'm right-handed and right-eye dominant. I injured my right hand a few years back and couldn't use it for shooting for about seven months, I was pretty seriously into IDPA at the time, so I taught myself to shoot left-handed and with my left eye. I tried to use my right eye, but I just couldn't get the two to coordinate. The angle is just different enough to throw me off.
It took a lot of practice, but now I can shoot almost as well with my left hand/left eye as I can with my right hand/right eye. I also practiced transitions from right-hand grip to left-hand grip. (Also, if you know what you're doing, you don't need an ambi-safety on a 1911.) When I was shooting in IDPA, it help out my barricade shooting a lot.
How to do it? Probably a lot of different ways. What I did was aim at a target past an open doorway that only my left eye could see while holding the gun in my left hand. Once I could focus my left eye on the sights, I put a target on the other side that only my right eye could see while holding the gun in my right hand, and then I moved back and forth between the two while making my right grip/left grip transition. Never needed to close one eye or the other.
Starting out was very difficult, but once it started to work, my proficiency developed rapidly.
Some people would probably look at the process as a pain. I found it fun (after it started to work). I also like that I've developed a new skill. After all, could be that my right hand gets injured again, and that whatever caused the injury also damages my right eye. Or maybe I just injury my right eye somehow. If either of those things were to happen, they would no longer leave me at a disadvantage.