Prescribing antidepressants is just like prescribing any other drug. Must be done by a doctor. Any general family doctor can prescribe antidepressants, but many people see a psychiatrist (doctor specializing in mental stuff) for those prescriptions. Some psychiatrists also do therapy, but many therapists are either psychologists or social workers, and therefore cannot prescribe meds.
Just to clear something up...
"Therapist" is a generic word used to reference many types of clinicians. Psychiatrists, who are MDs, are sometimes called therapists and they can prescibe medication.
Other "therapists" include clinical psychologists (phd, psyd), social workers (licsw, sw, counselor) etc.--in most states they cannot prescribe medication; however, most offer referral services so you can see an MD (psychiatrist, general practioner etc) for short appts and do your therapy with someone else.
Hope this helps!