when i first started climbing, my older brother, a long time climber always had his saw on the left.
i thought it didn't make sense if you are right handed, so I changed my first saddle to carry the saw on the right.
this way i can hang up my saw with my right hand and do it quickly, a frequent habbit i have is tossing the saw in the air slightly, catching the saw lanyard at the ring and hanging it up. I do it without thinking about it and it's very fast I think.
I'm trying to get my team-mate climber to change his saw from his left side to the right side to speed efficiency. but once your used to something, you don't want to go through the changing period, i understand that.