My personal experience is that it's better to shoot both LH and RH. I am right eye dominant, but have become almost as proficient shooting LH. It seems like such a high mountain to climb to shoot with the other hand, until you try it and find out it's just a small hill, and a fun one to climb at that.
I will probably continue to teach students based on their dominant eyes, or whichever side they want to use, but in the future I will demonstrate both sides, and advise them to learn to shoot from both sides, both for physical fitness and confidence purposes.