It's essential to have the paddle across the boat so that when your bum lands in the seat you can instantly have the paddle in both hands ready to brace.
Leaving the paddle on one side means that there's a crucial instant when you're in the seat, but you have to lift the paddle up and swing it across... and all kinds of things can happen in that instant that will cause you to be back in the water.
See also:
Remounting
It's getting a bit dated now and we need to renew it with some video, but it gives you the theoretical idea.
Rob
Currently Epic V10 Elite, Epic V10 Double.
Previously: Swordfish S, Evo II, Carbonology Zest, Fenn Swordfish, Epic V10, Fenn Elite, Red7 Surf70 Pro, Epic V10 Sport, Genius Blu, Kayak Centre Zeplin, Fenn Mako6, Custom Kayaks ICON, Brian's Kayaks Molokai, Brian's Kayaks Wedge and several others...