can you bring your own dog with you on a house sit?

I say it really depends on your dog and the client's dog. If I know my dog and the client's dog would have a great time and the whole stay would be like cousins visiting for a few days, I might ask the owner if they would mind. But if I have any doubts about the dogs getting along or enjoying being together or if I wasnt sure about my own dog's manners, house breaking, etc, I wouldnt even consider it.

No one here can speak for all owners. You would have to ask your own customer for permission to do so.

However, it would not be a good idea. Way too many liability issues.

I would never even consider bringing my own dog to a clients home

No unless specified by the owner with consent.