Psychologists do not. Psychology is a "soft science" in the same realm as anthropology, polisci, and sociology. Licensing varies, but typically requires a PhD, PsyD, or at least MA, for clinical practice. This doesn't preclude those with less education from acting as "counselors", however. Like most of the soft sciences, there is no established single or primary modus, and many different more-or-less competing theories and models exist, with various degrees of support and adherence. And like other soft sciences, is far more prone to woo than the hard science are; though as many at JREF have noted, the hard sciences are not entirely free of woo either. The less precise the science, the more woo there is.
"Therapists" or "Counselors" don't necessarily require any training at all. Depending on the state and field of practices, the typical therapist will have a Masters, but not necessarily in Clinical Psychology per se; but possibly in "Social Work", "Behavioural Counselling", or even "Divinity", as long as a certain amount of CP training is part of the degree (requirements vary). No medical background is required at all.
Psychoanalysis is not, as many people think, a sub/superset of psychiatry, nor is it related; but is rather an outdated school of psychology/psychotherapy. Although in the US admission to psychoanalytic training has been mostly restricted to MDs, that has loosened up a lot in recent decades, and many are from non-medical fields. In Europe and Latin America, there has never been any such restriction, and many more psychoanalysts are from non-medical backgrounds.
It is a tempermant thing many psych majors and social workers are hard knock sceptics. But therapists, well, many have woo as thier middle name.
