Rhinitis means inflammation of the nose, which usually leads to a lot of discharge. In dogs, this can happen for a number of reasons. Some of those reasons, such as parvovirus or a bacterial infection, are contagious to other dogs.
According to Ned F. Kuehn, DVM, the most common causes of chronic canine rhinitis are fungal rhinitis (a fungal infection in the nose), a tumour in the nose or lymphoplasmacytic rhinitis, which is a chronic nasal disease. None of these are contagious to other dogs.
One uncommon cause of chronic rhinitis in a dog is an infestation of nasal mites, which is contagious to other dogs.
Other bacterial infections, such as canine distemper, that a dog's suffering from can cause chronic canine rhinitis. These infections are contagious to other dogs.
Until you properly diagnose the cause of chronic rhinitis in a dog, isolate the dog from other dogs. Wash your hands before touching a healthy dog.
According to "Dog Owner's Home Veterinary Handbook;" Debra M. Eldredge, DVM, et al.; 2007, the prognosis for curing chronic canine rhinitis isn't good. But, if the cause of inflammation isn't contagious, the dog can be with other dogs.