Castile soap is supposed to be very gentle - it is an "all-purpose" soap, which is often highly recommended for washing very delicate textiles, as well as leather and suede. It is gentle but strong - you do not need very much and it doesn't lather very much . Here's a list of Dr. Bronner's Peppermint ingredients: Water, Organic Coconut Oil, Potassium Hydroxide, Organic Olive Oil, Mentha Arvensis (wild mint), Organic Hemp Oil, Organic Jojoba Oil, Organic Peppermint Oil, Citric Acid, Tocopherol (which is essentially Vitamin E). There are several other varieties as well (including "baby mild") if there is concern that the mint/peppermint might cause irritation.I would not use any shampoo that is not specifically for pets. Too many variables, and with all the pet products out there, not worth risking an irritation IMO.
Dr. Bronner's has similar ingredients as this "all-natural" shampoo: http://www.royalpetclub.com/Grooming/Shampoos/13-5-oz-Organic-Almond-and-Shea-Butter-Shampoo.html. The potassium hydroxide in Dr. Bronners is an ingredient in the process of soap-making, and none remains in the final product. Chances are there are similar caustic ingredients used in the production of other dog soaps/shampoos which may or may not also be listed on the label. That is how soap is made - you take a fat and add a very strong base and chemistry happens.