One of my neighbors has a cat which eats these anoles.
http://en.wikipedia.org/wiki/Polychrotidae http://en.wikipedia.org/wiki/Gecko
My dog just hunts them after discovering that they do not taste all that good.
Think this depends on the pet and what kind of interactions it has had with these lizards.
The neighbor's cat hat to be rushed to the vet because it got stuck in her throat. She thankfully got through this experience OK but still goes after the lizard that encroaches on her territory.