Shampoos for whitening are used to keep, clean, and preserve a healthy coat. In addition, this whitening recipe can repair your dog's white coat to a brilliant sheen without the use of harsh chemicals such as bleach.
So there are no side effects to this shampoo unless your dog has unique allergies to some of the products.
The skin of some dogs, for example, may not respond well to dish soap in general. You should be able to determine this if you just wash your dog once with a mixture of dish soap.
Being aware of your dog's allergies will be the best way to determine how your dog will react to this recipe. That said, nothing in this recipe is a harmful or toxic chemical.