"Zoonosis" is the process of transmitting a disease from an infected animal to a human. This suggests that animal-to-human transmission is not common. HIV is believed to have first spread to humans zoonotically, but has since mutated to a human-only strain.
This all raises the question: Why do diseases mostly just target a particular species? Aren't most metabolic processes very similar across species, especially within the same order and class? Dogs and cats can't catch the common cold, but they have similar diseases of their own.
There are obviously enough similarities that mice can be used in preliminary tests of human drugs, but the results are very tentative until human trials are done.