Are acupuncturists doctors or therapists?

Depends. This depends on the individual acupuncturist. Some are doctors that have received additional training in acupuncture. Others are specially trained "therapists".
Healers. Acupuncturists may be do's, md's or doctors of traditional chinese medicine. In asia - often one member of the home treats all the members of the extended family with acupuncture. This person may have been formally taught acupuncture or may have been trained by a family member. This is very helpful in areas where there are not enough doctors to go around. So it depends.