In any martial arts, there should never be a 'weak side' and a 'strong side.' Same goes for fighting with a pistol. We should have a left side and a right side and it should be our goal to be as proficient as we can be with both. I shoot a lot with both hands / both eyes. Its difficult for a while to learn to use both eyes and keep them both open but practice allows you to use either eye to aim the pistol.
For certain types of movements, it would be very advantageous to be able to shoot with either hand. Also, in force on force training, most hits are made to the hands and arms. If we're proficient with both hands, then we can fight more effectively, even if injured.