As Lacey came from a chihuahua breeder to me at nearly 2 years old and had never been left "alone," she suffered from separation anxiety. Every time I wasn't in her sight/range of hearing, she would howl (there's several posts on here about it!). She was driving me crazy, so I decided I was going to try fostering a pet to see if an anima companion helped her. Fostering a kitten allowed Lacey the company she was seeking, and her howling stopped from day 1 with her new friend. Afraid her separation anxiety returning, and too kind to take her new best friend away, I ended up letting Lacey adopt "her kitten."
I'm sure since the chihuahua is a pretty social dog and very reliant on the company of others that they benefit from companionship when you are not around. After seeing the change in Lacey's overall attitude since getting her kitten, I would definitely recommend having more than one pet. It doesn't even have to be another dog