All mine have had doors on the passenger side. I have seen them with doors on the left side, but the ones I have seen usually have living quarters for that door.
Maybe cause most folks are right handed? Heck, I don't know.
I have had a trailer with the door on both sides.
Though I find them more commonly on the passenger side I prefer the tack room door be on the driver's side. If only carrying one horse it rides on the driver's side so it is easier to have the tack room there.
I'm not experienced with horse trailer, but I am with regular RV's. If you are at a horse motel or campsite w/horses, your neighbor's door most likely will be on the passengers side. So if your door is on the driver's side, you will be facing your neighbor and will not have any privacy, nor will he. Just a thought.