The common cold (upper respiratory infection) in people is actually caused by many viruses. In general, the viral symptoms run their course and do not require anything more than optional symptomatic medication, unless complicated by a secondary bacterial invasion. Colds are usually of little concern unless the patient is less resistant to illness. Animals, as people, can get viral upper respiratory infections. Many of these can be prevented by making sure your pet is vaccinated every year.
Colds caused by viruses are not transmittable to other species, because upper respiratory viruses have generally evolved to affect specific species. Bacteria are somewhat less “choosy’ about where they settle. Bacterial respiratory infections could be, in theory, passed between people and pets, but the chance of this actually occurring is minimal in healthy individuals.