We had this happen once and only became aware because of our RING doorbell (the sitter came out countless times after midnight calling for the cat.)
I would expect the sitter to contact us immediately (this one did not) and work with us on a plan. In our case, we reached out, wrote a Facebook post for our local community and suggested she open the garage door and place food there, since our cat hangs out occasionally there during the day.
The plan worked. She was missing for 2.5 days and finally came home tired and hungry at 2 am. She was filthy, but we were relieved to get her home.
Bottom line: The sitter should contact the Owner ASAP if an animal goes missing, since the Owner will have the best intuition about how to get them to return home.