I continually wore my medicine bag on my right arm... but then I got a Dior saddlebag... which, if you ever pick one up, you will notice that it is designed to go on the left shoulder. At least, I think it is. I just got my LV bag with two handles but I haven't carried it around enough to see if I kept my left hand habit (which is actually a cultivated thing)
so, yeah, right handed. Left shoulder, right elbow. (not at the same time)