'the dogs are staying with us"
She doesn't want the dogs to be taken away. She says, they're staying with us, as in, we're keeping them.
4 Answers
Los perros se quedan con nosotros
Los perros se quedarán con nosotros. (If you're refusing to give them up / turn them over to someone else.)
"Los perros se están quedando con nosotros"
Estamos de mantenimiento de los perros.