on (one's) watch
Under one's supervision; while one is in a position of power, authority, or responsibility. I can't believe you let this happen on your watch. It's strange—even though it wasn't on my watch, I still feel partly responsible for the robbery.
on someone's watch
while someone is on duty; while someone is supposed to be in charge of a situation. I am not responsible since it didn't happen on my watch. I guess I have to bear the blame since it happened on my watch.