I shoot lefty and righty. Shot right handed with a dominant left eye for better than 20 years. Went to lefty in 2005. It took two weeks to learn to shoot lefty, it's no big deal...if you don't fight it. Feels awkward and it isn't sub concious, but making the switch isn't all that hard.
I don't know if there is any actual benefit while hunting. Out of a stand you always have blind side and being able to shoot with the other hand makes sense. What you have to account for is time and movement and it has been my experience that by the time you know for certain which way the critter is headed it is already to late to make the switch. That said sometimes stand placement is a better criteria for the shooting hand preference.
I've had more benefit at the target range. If you can shoot both ways you stand the opportunity to clean up in a shooting competition, just be creative...8^)