Right handed - on my right side (strap across my body onto my left shoulder if it’s a messenger bag, or on my right shoulder if I’m carrying a backpack as a shoulder bag
Wow, weird! I’m like the OP; I totally expected the results to end up as purse being on the non-dominant side.
I’m right handed and I hate hate hate my bag on my right side.
Left handed, bag on right side. Anything else would feel weird.
Right handed, prefer to carry a shoulder bag on the right side.
As somebody already noted, that leaves my “smart” hand free to get something out of the bag; many bags have extra pockets on the exterior. Bags with double straps (1) can be opened while hanging and (2) stay up better than single-strapped bags.
See, my reasoning is exactly the opposite. I have my bag on my left shoulder so my dominant hand can be free while the left hand roots around for whatever I need.
So do I. I’m right-handed, and it just feels weird to carry a bag on my right shoulder. It falls off rather quickly.
Right-handed and switch it back and forth, but probably carry it more on the left shoulder so that my right hand and arm are free.
Lefty trained as righty, and the bags change sides. If I’m only carrying the bag, it changes more-or-less periodically; if I’m carrying something else, it depends on what and what for.
I prefer to carry my man-purse Jack-Bauer style, bag on my right, but slung so the strap’s on my left shoulder.
I fit your hypothesis as well, and am a right hander with bag on the left side.
Mostly left-handed, carry on my right. I was somewhat ambidextrous as a young child. Wrote with both hands. In I think first grade, the school people decided I had to use just one hand… it ended up my left. However, I do a lot of things right-handed, probably because my brain couldn’t decide which hand should be dominant. Also, a lot of utensils in my house were right-handed, like scissors. I carried my baby (many years ago), on my right hip, but sometimes switched when my arm got tired. I bat and catch lefthanded, but throw with either (and poorly), I cannot for the life of me use chopsticks with either hand 
This. I don’t have kids, but whenever I’d watch my sister’s kids, universally I would pick them up and carry them on my left hip, and only switch if my arm got tired. I’m also right handed and carry my purse/messenger bag/bookbag on my right side.
Nope, not at all. I usually hold the bag open with my right hand and dig for stuff with my left. When we did the left-handed/right-handed tests in a biology class ages ago I came out to be slightly more right-handed, so perhaps I’m closer to ambidextrous? No idea.
Also, I almost always wear my watch on my right hand and am told this is really weird. It does annoy me when I’m writing so I don’t wear one often, but it feels weird on the left side…
I wasn’t sure whether to put right-handed or ambidextrous (I eat with my left hand and generally favor it for heavy lifting or any task I can use either hand for, but can’t write very well with it). But I answered I carry my purse on my left shoulder. That’s what feels most natural, don’t know whether that’s from force of habit due to my left-side favoritism, or that I have a natural inclination to do so.
Anyway, one of the right-handed left-bagged votes is mine. 
Shoulder bags go on the left. Messenger bags with straps that cross the chest also usually go on the left shoulder, which means the bag ends on my right hip. I’m not fanatical about it though, it will vary depending on what else I’m carrying.
Oh I guess if people are putting their reasons for which side they put the bag…
I always want bags on the right, even if I’m carrying them in my hand. If I need to use my right hand, I’ll put down my bag. Having them on my right gives me better control of the bag while moving or riding a moped. I never open the bag or reach into the bag while moving, I usually stop, take off the bag, then open it (my only shoulder strap bag is my laptop.)
Right-handed, and I carry it on the right. If it’s got a long enough strap, the strap goes over the left shoulder and the bag rests against my right hip. Any other location just feels weird.
Similarly when I sling my computer backpack over one shoulder, it’s always the right shoulder.
Right handed, but switch the bag from one side to another. If you keep it on one shoulder all day, it starts to get sore. It just makes sense to redistribute the load when it begins to feel heavy.
Right handed, right side. I have no idea why.
Right handed, right side. Moreso the past nine weeks, what with a cast on my left arm.
Right handed, and use a cross-torso bag that rests on my right hip. If it was slung so it rested on the left, I’d have to constantly pull it over to reach in with my right hand because that would be more comfortable. This way, I can reach all the way to the bottom with no problem or re-positioning and my left hand is free to do less dexterous things like opening doors.