The simple answer is yes. As part of your training you will have encountered COSHH (Control of Substances Hazardous to Health) which requires you to wear PPE. Face masks are not the only form of PPE; gloves, aprons, goggles and face shields all help to protect you.
The requirements on what PPE you need to wear is different for each salon and technician. This is why it is important to perform a risk assessment for each business.