I'm left handed.
Growing up in the 50's and 60's, it was a right handed world. The only thing I can't do right handed is to write and throw balls. I bat right handed. I was taught to shoot right handed. I am right eye dominant...
But after I got out of the Marine Corps, I slowly learned to shoot left handed (rifles only). When I finally broke down and got some handguns, I started shooting with both hands. I figured, what the heck! No use in trying to learn by using just one hand...
It simply takes practice. Shoot a few sets right handed, then switch. After a while, it will become natural. The benefits are of course, real. You never know how a situation will come about that might preclude your shooting with your dominant hand.