Do psychiatrists have educational requirements other than an MD?
The only required degree to become a psychiatrist is a doctoral degree (M.D.). Using your example, yes those people could train to become psychiatrists.

After passing the Medical College Admission Test (MCAT) and going through med school, most students undergo a psychiatric residency (aka. clerkship) where they are trained in different medical centers (hospitals, health clinics, psychiatric wards, etc.). This allows the prospective psychiatrist to gain "real world" experience. After a psychiatric residency, psychiatrists must then take an exam issued by the American Board of Psychiatry and Neurology. If they pass this exam they are certified and can then legally practice as a psychiatrist.

Source:  http://study.com/articles/Psychiatrist_Educational_Requirements_for_Psychiatrists.html

Edit: Med school teaches you theoretical psychiatric knowledge whereas the clerkship will teach you how to apply it. The clerkship could also train you in specialty fields.