With all the birds I have worked with I have never had one "cut" my hair. Many birds will preen your hair some will even pluck a few strands now and then but they don't cut it off. But yes I can just imagine a person with an affectionate bird having a bald head on one side LOL!!
Also with a new bird it is not a good idea to have them on your shoulder. Some people say it is never a good idea to have a "shoulder" bird as they can do a lot of damage to your face with their beak if ever startled. I do have shoulder birds and always have but not until I have had them for quite a while and they are very well trained so that I trust them. Even then I try not to get them in a situation where they can be easily startled while on my shoulder.