There are two different photo sections in Rover. Your profile photos, which usually incude photos of you, your home, your pets, etc that you upload manually, and your Canine Clients photos, which consist of photos you send to owners through Rover while watching their pet.
As Walt said, you can decide which pictures to make public under Canine Clients by either turning the feature off completely under Settings>General, or case-by-case when you send each photo from the Rover app - there is a toggle switch at the top that says "Make Photos Public".
Owners also have the option to switch off making their pets' photos public. I assume this is something they can do in their settings as well, so that no matter what you select their photos won't show up on your Canine Clients or under their pet's profile. I've never asked a pet parents permission before sending photos, and I automatically post most photos to my Canine Clients section. If a pet parent does not want their pet's photo published, they can control that on their end.
However if you have a business page or create marketing material, I would request an owner's permission before using their photos. I also don't upload pictures of other people's pets under my profile photos, but think if you choose to do so getting the owner's permission first would be polite.
Lastly, if you are in the client's home, be aware of what you are posting publicly. You don't want to embarrass an owner with a picture of their dog with a dirty kitchen in the background, or include photos of items the owner may not want splashed on the internet (expensive TVs, paintings, etc.)