Traditionally buggy drivers sat on the right because most were right handed and the whip could be wielded without possibly hurting the person seated next to you. Also, in times when one needed to have his gun hand handy it was more convenient to be sitting on the right hand side.
Seating the driver on the left side began in America with the advent of the automobile. The earliest automobiles had steering wheels on the right side but that changed in 1908 when Henry Ford’s Model T put the steering wheel on the left hand side. His rationale was that the driver could see the center line much easier and could also gauge the distance between an auto coming in the opposite direction much better.
I might add that today only about a quarter of the world drives on the left side of the road. That quirk goes back to the days when the sun never sat on the British Empire and they mounted their steering wheels on the right (er, wrong) side.
Riders, especially strangers meeting on the trail or on a road would most likely pass each other on their left. That kept the right hand or gun hand free to deal with any funny stuff.