I have really sensitive skin, and am prone to reactions to a lot of chemicals. But I have not had any problems washing my hands with it. That said, I would be hesitant to use it in a place that was already irritated or prone to being very dry.
Dr. Bronners is more natural, but KMF is still so much better than traditional soaps. Some of Dr. Bronners scents are not pleasant to my family, so when we use that we get unscented. They are more moisturizing than Dr. Bronners.
I, like the others, definitely think it's a user preference in this case. I always say to do your homework especially if you're looking to find the best products for your family. I love the Kiss My Face soap though!
I agree with Julie, it depends on what you are looking for. Kiss my face attempts to use ingredients that are better for the earth but B+Dr. Bronners has fewer ingredients and better for you and the environment.