I use nair sensitive hair removal cream, but have not received any negative side effects yet. Only if I leave it on for more than 10 minutes it then starts to sting slightly. Also, the aim is to pat the area with the cream and not to massage it in. 
Sting slightly, yes that's because it's starting to react with and dissolve your skin just like it's dissolving the hair.
Remember different people have different sensitivities regarding things like skin and their reaction to chemicals and even sunlight etc
AI Overview of the active ingredient (Of the one I knew about)
Thioglycolic acid (TGA) is a colorless, highly corrosive liquid with a strong, unpleasant smell, containing both thiol and carboxylic acid functional groups. Primarily used in cosmetics for hair perming and depilatory creams, it is also used in PVC production, water treatment, and pharmaceutical manufacturing.
TGA is highly toxic by inhalation or skin contact, requiring strict handling protocols.
Key Aspects of Thioglycolic Acid:
Formula & Structure:
(also known as mercaptoacetic acid).
Hazards: Listed as a hazardous substance (CAS 68-11-1). It causes severe skin burns and eye damage, and is toxic if swallowed or inhaled.
Applications: It is the active agent in hair straightening and waving kits (perms) and hair removal products. It acts as a reducing agent in organic synthesis, metal finishing, and the production of stabilizers for PVC.
Safety Measures: In case of exposure, immediately flush skin with water and remove contaminated clothing. Use only in well-ventilated areas, as it can cause sensitization.