What kinds of specialists diagnose and treat fibromyalgia?

Team effort. Fibromyalgia appears to respond to multi-modal treatment. Doctors from many specialties often work together: rheumatology, psychiatry, pain, family, etc.
All. All doctors can diagnose because criteria are well defined & specific. Latest criteria is a short questionnaire. Previous criteria included physical finding of pain points, but theses no longer required. This gives insight that cause is unknown & criteria for diagnosis is similarly shifty. On my chronic pain board at quora.Com, i present a full discussion of the "true cause of fibromyalgia".