As a site owner, you stay in the middle of the doctors and patients. Let the both parties be together for treatment and consultation purposes. Any patient or anyone on behalf of the patient can book an appointment via a form on the appointment page. The appointment page takes some of the patient information in order to contact the patient in future. Below screenshot is from Shaper Medico Template.

Appointment Form is SP Medico component view, all messages sent by patients are stored in component admin area.