Yes to both! (Depending on circumstances) No runner on base, catcher should conserve energy and stay low. Hand stays behind leg (Preferred) or back until pitched ball is in glove. Runner on base catcher rise in stance (runner on) with hand behind glove. Hand stays behind glove down (Pitch down the middle) or buries hand to chest protector on inside /outside pitches (Hand tends to separate)
Below is what we typically see catchers do with runners on base and hand behind the back, leg or set beside the leg... (click to enlarge)
The catcher will anticipate the throw and bring the hand out next to the glove so as to be ready to catch and throw. (Similar to an infielder catching and throwing) THIS is when too many catchers get broken bones or torn ligaments.
Below is an ACC 10U catcher demonstrating her runner on stance with the throwing hand behind the glove.
Her throwing hand stays behind the glove and is ready to transfer the ball to throw. We challenge our catchers that after the pitch sign is given, DO NOT LET YOUR PITCHER SEE YOUR THROWING HAND UNTIL THE PITCH IS IN YOUR GLOVE OR HIT. If the pitcher can see it, more than likely the pitch can find it. It is exciting to see more and more catching coaches in softball and baseball recognizing the hazards to the throwing hand and teaching their catchers how to keep their hand safe. More and more are teaching their catchers to put the hand behind the glove.
Even with keeping the hand behind the leg, back or beside the leg with no runner on, catchers must train to keep the hand still and out of sight behind the leg or back. Just lifting it out a few inches invites injury.
Whatever you choose to do with your catcher... Please don't let their pitcher see their throwing hand.
-Chaz @ ACC