According to Dr. Justine A. Lee, DVM, DACVECC, writing in Pet Health Network, shop bought mushrooms, including Shitake mushrooms, are safe for dogs to eat.
The prime things to remember, however, is that we often cook mushrooms with various oils, sauces and other ingredients which can be harmful to your dog if these are fed.
You should only serve mushrooms that are plain or that have been cooked on their own with no added ingredients.