I learned a bit about natural deodorants when going through chemotherapy last fall. I was doing cold-capping, which is a procedure that can prevent noticeable hair loss during chemotherapy. Cold-capping patients have to follow strict protocols for it to work, including no aluminum deodorants/anti-perspirants. Apparently aluminum builds up in the hair follicles and can weaken them.
From some research, I learned that many natural deodorants will be irritating if they include baking soda. So, I've stuck with the "sensitive" versions of natural deodorants, which do not contain baking soda, and haven't had irritation. Used Schmitt's for a while (can't recommend it, caused a clumpy moist mess so I had to put on a "blotter" shirt before changing to my actual blouse for the day), but am now using Native's Sensitive product (aloe/green tea version), with very good results.
Sweating does happen but there is no odor (admittedly, there was a "transition period" as my body sweated out the chemicals). Although I'm done with cold-capping, I'm continuing with the natural deodorant because I'm concerned about the health effects of chemicals in regular deodorants/anti-perspirants. There are a lot of lymph nodes in the underarm area, and my goal is to keep them as healthy as possible.