House sitting usually means sleeping at the client's home.
As for bringing your own pup, you need to discuss this with the owners and have a plan for your own pup if the owners are not inclined to having your dog stay in their home.
I only ask if I feel that the dogs might get along. Based on my own dog, I can usually tell right away if it will be an unlikely pairing (my pup is high energy, bordering on annoying for older dogs).
If the owners agree, I try to have the dogs meet before the stay, just in case it ends up not being a good fit with my dog and I need to find other arrangements.
If all goes well, then I have my pup wear a belly band during the stay so that he's not tempted to mark. I call these his 'stay wear'. :)
Most owners don't like last minute surprises, so it's best to bring this up soon.
All the best!
Your house sitting rate is less than boarding or dog care. Staying at someone else's home should be considered a premium service since you are watching not only their pet but their home and possibly doing other tasks, like taking in mail, watering plants, etc.