You cannot effectively sanitize wood. It doesn't matter though, as the boil after the mash will kill off the baddies. There are probably more baddies on your grain than your mash paddle.
Use whatever mild, clean rinsing detergent you would use on your dishes. Avoid cleaners like PBW of Oxiclean, as they can sometimes stay in the pores of wood, even after multiple rinsing (ask me how I know)
A quick spritz of star san, or a stir in Iodaphor after a cleaning is all you need.