Cats probably transmit COVID-19 efficiently, but dogs less efficiently.  Both stray and domestic cats in Wuhan had antibodies to SARS-CoV-2 after the epidemic started, but not before, perhaps indicating infection from humans (Zhang Q et al.).  Experiments found that cats can be infected with SARS-CoV-2 by intranasal inoculation and by staying near an infected cat, indicating respiratory droplet transmission.  Since stray cats typically interact minimally and indirectly with humans, this shows how transmissible this virus is.  Similar experiments with intranasal inoculation found dogs transmitted less efficiently than cats (Shi et al.).  A Hong Kong dog and a US dog were infected, perhaps by their owners (Office International des Épizooties; WRAL).  Two cats in New York state were found infected (Centers for Disease Control and Prevention 2020c; Peltz).  These studies indicate that household mammals could function as a vector and reservoir of SARS-CoV-2.  

The US CDC recommends that if people and animals live together, both animals and people should similarly take precautions and seek treatment (Centers for Disease Control and Prevention 2020c; 2020d; 2020e):

Do not let pets interact with people or other animals outside the household.

Keep cats indoors when possible to prevent them from interacting with other animals or people.

Walk dogs on a leash, maintaining at least 6 feet from other people and animals.

Avoid dog parks or public places where a large number of people and dogs gather.

If  a person or animal is sick with COVID-19 (either suspected or confirmed by a test), restrict contact and stay isolated from  other people and animals.

If a person or animal goes to a doctor, nurse, veterinarian, etc, tell the health professional about any exposed or sick people or animals in your home.