I wouldn't want someone without a medical degree prescribing me drugs or recommending treatment. Ofc you need a medical degree.
Studying psychology allows you to become a psychotherapist or psychologist. Studying medicine allows you to become a psychiatrist.
Also, you should research the difference between both professions to see which one suits you. Psychiatrist tend to look at the neurological and physical side of mental illness; whereas a psychologist or psychotherapist looks at the cognitive and behavioural side. The latter is not permitted to prescribe.
Note: In order to become a psychiatrist, you will need to be a "normal doctor" for several years. This means A&E work and all the trappings of being a junior doctor. Then you have a few years of training to specialise as a psychiatrist.