i voted left side cos if i carry it on my arm i always do the left (i'm right handed)..... but if its on my shoulder (i dont own many shoulder bags), i tend to switch between both sides.
I carry most bags on the left - easier for me to reach across into them with my right hand while holding in place with the left. A clutch or wristlet I like to hold with right hand though.
I usually carry satchels that also have shoulder straps. When I'm carrying it on my elbow or in my hand, I usually have it on the left. When I carry it on my shoulder, it usually goes on the right shoulder.
i'm left handed in everything other than writing and carry my bags on my right side. it just feels more natural. it seems like a lot people who are right handed carry their bags on the left side. maybe it's the opposite side of what you are because you'll have the other hand/shoulder free to do things?