The answer: Generally, no. Research suggests that dogs do not catch human colds (aka rhinoviruses) because their genetics are not susceptible to the same strings of viruses as humans. However, dogs have their own virus strings, such as Kennel Cough—a highly contagious, cold-like dog virus that resonates high-density dog populations such as kennels and shelters—and even their own string of swine flu (H3N8).
Most diseases are species-specific, so dogs only spread germs among dogs, cats among cats, and humans among humans. There are rare exceptions, however, such as H1N1, which is a pig virus that mixed with a human virus and mutated into a human-to-human virus.
Though dogs and humans do not pass viruses back and forth to each other, they are susceptible to some of the same bacterial infections and parasites (ring worm, intestinal worm and rabies), which can sometimes lead to cold or flu-like symptoms. So it’s always good practice to wash your hands when cleaning up or dealing with animals.
Conclusion: You are probably safe if your dog sneaks in a lick on the face while either of you are suffering from a flu or cold.
“To date, there is no evidence of transmission of canine influenza virus from dogs to people and there has not been a single reported case of human infection with the canine influenza virus…Influenza viruses are constantly changing and it is possible for a virus to change so that it could infect humans and spread easily between humans”—The CDC Key Fact Sheet.