I am uncomfortable with mine on the right. Mine is always on the left unless I need to carry my crop on the left. I’ve honestly never paid too much attention other than when I see people with way too long of reins which is a noticeable distraction. Now that it has been brought up, though, I’m sure it is one of those things I’ll randomly notice now 
My first trainer was 75 years old and this was in the 80’s, so take it as you will:
He insisted the bight was always carried on the outside for a cleaner picture (flat classes usually had judges standing in the middle of the ring).
I think a few trainers later I learned “bight is on the right”. It’s always been a bit confusing, though.
Wow - it’s something I just do relexively, but sitting here in the chair (in another country), I can’t recall whether I’m a “bight on the right, always” person or not. 
I know I’m a leftie now, but looked at some older videos from high school to see if it has been that way a long time. Sure enough, every one I have mine on the left!
The common sense reason for the bight being on the right is to keep it out of the way and avoid the possibility of putting a foot in it while mounting, and dismounting.
In all my years of riding it has never gotten in the way of my whip.
I have done this in the past where you put the bite on the side away from the judge and have your whip on the side facing the judge. it can look messy with bite and whip on same side. I mostly adhere to bite on right.
Auto correct on bite!
Ah well, no real consensus then…
Personally, I don’t notice unless it annoys me, such as interfering with my whip or getting tangled in mane but that is because I always carry my whip on the inside unless dealling with a specific ‘outside’ issue.
I was taught that the bight was on the outside, and the crop on the inside. We were even taught how to move the crop smoothly to the other side when changing direction. I learned this 40 years ago. 
