What is the risk to an owner from an animal's malignant disease? At this time, there seems to be no risk to humans from animal cancer (fig. 1).
Figure 1: Animals with cancer pose no risk to humans.
Several approaches have been used to evaluate the possibility of transmission. First, animal and human cancers that occurred in the same household were studied to determine whether this simultaneous phenomenon occurred more often than would be expected due to mere chance. However, there was no increased incidence of cancer in humans who had lived closely with an animal with tumors. Secondly, no viruses that are known to cause cancer in animals infect humans, at least as far as we know. These viruses seem to be species-specific; that is, they can be spread from an infected animal only to another member of its own species and do not cross species lines. Thus, although our knowledge is limited, we currently know of no transmission of cancer from animals to humans---or from humans to animals, for that matter.
In cats and dogs, the most common "contagious" cancer we know of that can be spread from animal to animal is caused by the feline leukemia virus (FeLV). FeLV can cause lymphoma and several kinds of leukemia in affected animals and is transmitted between cats in saliva (fig.2).
Figure 2: Cat infected with feline leukemia virus. 
The only known contagious, virally-induced neoplasm in dogs is the papilloma (wart) commonly seen in young dogs and caused by the canine papillomavirus. Multiple, small, cauliflower-like growths may occur in the mouth or on the skin after infection with this virus, usually 4-8 months after exposure. As many as 50-100 tumors may be present at one time. The dog's body mounts an antibody response to this virus, however, and the tumors usually regress spontaneously in a few months. Transmissible venereal tumor is a tumor that can be spread from dog to dog by implantation of tumor cells onto a mucous membrane (vaginal lining, penile surface, oral cavity, rectal lining, etc.), but actual tumor cells are required---no virus has been implicated in this tumor's spread, and the tumor cannot be transmitted to any other species.
