It's on my right side in Sapphire, I believe it's on my left for Palinoia. Actually, I didn't know that it could be on different sides until I got a second copy of the game.
BOTH! In my NA town, it's on the right. In my JP town, it's on the left. I think the waterfall determines which side it's on, it seems to be the same side.
The right, I actually prefer the left side now I know about it.
When I started up my cycling town and went to check out main street I actually thought the museum had gone missing. Obviously I found it on the left in the end actualky every town I have been to so far has the museum on the left, my sons included