I advertise as a traveling sitter as well as hosting dogs in my home. I've only gotten one request to stay overnight in a client's home, and that was from a regular client who had an unusual situation with her dog. My main responsibility is to care for her dog (which is relatively minor), and then I'm free to do as I please. I do stay overnight but I'm free to leave for short periods of time to run errands and take care of other clients.
The rest of my visiting sitter bookings have been for 2-3 visits per day. My visits include walking, feeding, and play time, in general following the dog's usual routine. And I offer to bring in the mail, water plants, and turn lights on and off for no additional charge. If I were asked to stay overnight, I would not have a problem with it as long as it were not for an extended period of time and if I were allowed to leave on occasion (run errands, check my house, etc).
It is mandatory in my opinion that the specifics of any traveling sitter stay should be discussed at the meet and greet. First and foremost, you need to determine the pet's routine and how often and when to visit. If you will be staying overnight, you need to determine where you will sleep, TV and wifi privileges, kitchen access, and what else you might be expected to do in addition to pet duties. I charge my regular in-home boarding rate if I stay overnight, and I have a lower rate for daily visits.