There has been some controversy about whether or not that dogs have self-awareness. For example, they classically fail the self-recognition mirror test where you surreptitiously place a dab of make-up on their face and see if they respond to it when they look in the mirror. Human infants typically respond to this test by touching the mark of their own face from around about 18 months of age. Other species including chimpanzees also have self-recognition on the mirror test thought there is still some debate about whether all other primates have this ability. This by the way was one of the first disputed research studies that got my former colleague Marc Hauser at Harvard into so much trouble. He reported that cotton-top tamarrins recognized when their hair had been dyed pinkizzimo punk pink but later had to retract the paper.
One animal for which there has also been some debate is the domesticated dog. They do not pass the mirror self-recognition but many dog owners are convinced that their pets know what they are thinking. Dogs have been selectively bred from wolves over 15,000 years for their sociability but I am still not sure whether they can appreciate mental states of others. Without this ability, communication would be almost impossible as understanding another fully requires having a Theory of Mind.
Anyway, I have just received this evidence that settles the matter. Clearly they do.