Walk into psychiatrist Drew Ramsey's office in Manhattan and you'll likely be greeted by Gus, a four-year-old shih tzu. After escorting you through the waiting room, he may hop onto the ottoman and go to sleep or sit beside you on the couch.

Some patients pat Gus while they talk to Dr. Ramsey. A few talk to Gus instead. And if they get emotional, Gus provides physical comfort that therapists can't offer. "We can't hug patients, but patients can hug Gus," says Dr. Ramsey, who began bringing his dog to his office two years...