It's really no different from if you board dogs in your home. You can come and go, so long as you let the customer know that you do this and that the dog will not be left alone for longer than x number of hours. They need to know and agree to your plans.
House sitting is also like boarding in that it is supposed to be for 24 hour intervals. Moreover, it should be considered a premium service because you are not only taking care of the customer's pet but also watching their home and probably being asked to do little jobs around the house, like watering plants, taking in mail, etc. Therefore, the sitter needs to stay overnight in the home and be there a majority of the day as well. If you have other clients for dog walking or drop-ins, then let the house-sitting customer know that you do have other commitments but will be in their home a majority of the time and their dog will not be left alone for whatever time you need.
There might be some customers who require you stay in their home 24/7. If you want to accept that arrangement, you should be charging accordingly because you cannot do other jobs concurrently.
Finally, since you are an independent contractor you can decide for yourself how you want to handle house-sitting jobs. From what I read here, there are many variations and interpretations. Just be sure the customer understands the arrangement.