Did they have small breeds like we have today? I've read that British dogs were sought after for their assistance in hunting, and I know that Rome used dogs in the military, but I haven't seen anything about civilians having dogs for nonmartial reasons.
Yes, Romans certainly had dogs, there is even archaelogical finds of "beware of the dog" signs such as this one found at Pompeii , so they were almost certainly kept as pets/guard dogs, much as they are today.
The breed distinctly used by the military was an extinct breed referred to as a Molossus
