Mushrooms Are Good For You Both
Mushrooms are known to have a variety of health benefits, and some of them are good for dogs too. In general mushrooms support the immune system, supplement trace minerals, and contain anti-oxidants (including L-ergoothioneine which is not destroyed by cooking). More specifically different kinds of mushrooms offer more targeted health support.
Food vs. Supplements
In general medicinal foods are the way to go, unless there is a specific health concern. When food is consumed in its entirety, the body is able to digest and use it. It is generally safe to consume an amount that is physically possible to eat. Supplements usually contain more of a substance in a concentrated form, so they can be more effective to treat a specific condition. Mushroom supplements are on the safe side but it is always best to use supplements under the guidance of a health care practitioner like your doctor or vet.
Cooked vs. Raw
Although certain proteins and vitamins can be affected by cooking, they are typically denatured or broken apart, which is what digestion can also do. Generally the body can put these pieces back together, so don't be afraid to cook your health foods.
Dog Safe 'Shrooms
Shittake mushrooms have been used medicinally in Asia for 6,000 years, and are the powerhouse variety. They have been shown to contain compounds that can fight cancer, boost immune response to viruses, and aid digestion.
Button (which grow into crimini and portobello), reishi, maitake, chaga, shimeji, and turkey tail are also dog safe. These mushrooms can support immune function, cancer recovery including radiation side-effects, digestion, allergies, and arthritis.
For more info, check out this article from Animal Wellness Magazine.