In a full body massage given by a licensed massage therapist, you are fully draped with a sheet and no “private parts” are exposed at any time. You are left in private to remove as much clothing as you are comfortable with, get on the table and under the sheet then the therapist comes back in. The therapist will uncover the specific part to be worked on (arm, leg, back, etc) one area at a time and then redrape that area and move on. When you turn over, the therapist is holding the sheet so that you are not exposed to view.
You can also have a seated chair massage which is done fully clothed.