I've always trailered just my 1 horse in her 2 horse trailer, always put her on the driver's side for the same reason someone already mentioned above, because of how the roads are higher in the center and decline towards the right side (for water runoff).
I have never noticed a difference, but then, I am also a careful trailer driver in general and always make sure I take my turns slowly. My trailer has never so much as swayed with just my 1 horse in it, so I don't see it as a problem. But my horse is also a quiet trailer rider...she just stands there munching her hay, she doesn't kick or jump around or paw or anything.
"The times when you have seen only one set of footprints in the sand, is when I carried you..."