This depends largely on the airline, what specific mechanism got you on standby, and the mood of the gate agent.
A friend of mine had a free stand-by for life from United. These tickets have the lowest priority and frequently you basically wait outside the gate until they are ready to close the door and then they decide to let you on or not. Checked luggage was either not allowed (he never took any) or significantly reduced your chances, since it's extra hassle for the gate agent and ground crew.
I've done a few same day flight changes (mostly to get on an earlier connection) using Platinum or 1k status with United. First questions I always got was "do you have checked luggage". I'm quite sure that wouldn't have gotten a seat if I had checked luggage.
I'm fairly sure that the gate agent who manages the stand by list has some guidelines but also a fair bit of discretion. Criteria are booking class, fare class & rule and $ paid, status, reason for standy, culpability by the airline (delayed connection, overbooked, cancellation, etc.) and, yes, checked luggage, i.e. "how quickly can I get this show on the road"