It's best to put the weight on the bike instead of your body, if you can... Makes for a much more comfortable ride, and a lot less wear & tear on the rider.

I usually use a pair of Ortliebs, and try to balance the load. I'm a left-foot-downer, so if there's only one pannier (occasionally, and I have a little briefcase that clips to my rack) it goes on the left side.