Why would you want to?
IMO.. It's not a short board thing..
Do a race or downwinder with a side wind and you'll be paddling only on one side.. 
Paddling flat water using only one side is good for paddling in a big circle.. 
I generally find that if I've come off the back of a wave and I'm paddling back out to the line-up, I'm still in sort of a half surf stance with my weight over my toes, this also helps me to control my board a little more as it tends to want to climb over the foam rather than punch through it, in doing so my rail is somewhat acting as counter direction to my paddle stroke, allowing me to predominately paddle on my right side, I may have to take one or two strokes on the left to re-adjust particularly if I’m hit with a bit of white water knocking me off my line. I do the same thing if I’m surfing my 10’6 so not sure it restricted to ‘short’ boards. Can’t comment on flat-water paddling a 14 footer I gave that up a while ago.
Place the blade in the water so it is at 90 degrees to a line pointing from your feet to where you want to go to, rather than placing the paddle at 90 degrees to where the nose is already pointing. Then just don't change hands. A decent paddler should easily be able to paddle a mile or two on one side only - at least in mild conditions, anyway. It's a good technique drill, since it makes you think very carefully about tracking. DJ is right, when downwinding you may have to paddle many miles on one side only. But that situation is a bit different...
There is a very early video from Pono Bill (or maybe Bill Boyum) of Standup Zone fame about this, involving combining a modified sweep (like in canoeing) and weighting the opposing rail. Can't find the video, it's on a lake with houses and he's wearing a dry suit (IIRC).
pretty similar to what Area10 said, but basically what is called a "Bow Draw" in kayak/canoe where you enter the paddle at the start of the stroke a little further out than you normally would and have the paddle power face angled a little towards the boat (position will be pretty similar to what Area10 said). now start the stroke by pulling the blade towards the rail (this will pull the nose of the board back into line) once you get to your normal starting position straighten the blade and pull back as per normal. When I surf I spend probably +80% of me time on one side and only change for the sake of it as using this stroke you can manoeuvre in any direction with practice.
