Stephano and Caliban. In the play these mutts were said to have been "... divers Spirits in shape of dogs and hounds..."
Another unsolved mystery on that continent is a creature that is known for sucking blood.
No not Vampires, and especially NOT the sparkling ones.
Now I know what you're thinking. What does a Chupacabra have to do with the dogs that I mentioned earlier? Rest assured my friend, it will all make sense in the end. Knowing Shakespeare, he tends to have minor connections in his play, especially with Greek Mythology.
The reason I bring up Greek Mythology in this is because of the "hounds" that were talked about in the play. Now think of a mythical creature that is a type of hound. I'll give you a hint. They're found in Disney's Hercules and Harry Potter... Yes you got it! Hellhounds!
It would make sense that these dogs are Hellhounds because of not only Shakespeare's knowledge in Greek Mythology, but also the association he demonstrated with Prospero's sorcery and dealings with various spirits outside of the earthly realm.
Hellhounds are described as being paranormal guard dogs of the entrance to the afterlife. In folklore it is said that they have fire-based ability.
Recently it has been made clear that the culprit was in fact a type of dog that had mange. Mange is a skin disease found on mammals that is caused by parasitic mites which generally resulted in making their skin very itchy. Due to the over excessive scratching, the animal would lose hair and their skin would become "inflamed" and raw. Did you catch the connection there?
In conclusion the dogs and hounds, that Prospero owned, were Hellhounds that are now known as Chupacabras! I guess that answers the question "Who let the dogs out?"