OALD:
auto•didact noun (formal) a person who has taught himself or herself sth rather than having lessons

self-'taught adj. having learned sth by reading books, etc., rather than by sb teaching you: a self-taught artist

M-W:
self–taught
1 : having knowledge or skills acquired by one's own efforts without formal instruction a self–taught musician
2 : learned by oneself self–taught knowledge

AHD:
autodidact
NOUN: A self-taught person.

self-taught
ADJECTIVE: Having taught oneself without formal instruction or the help of others.

CALD:
self-taught
adjective [not gradable]
(of a skill or knowledge) learned or trained by yourself
a self-taught artist/musician

'Autodidakt' is a noun whereas 'self-taught' is an adjective. OALD's tagging as formal is the only evidence that 'autodidact' is rare. In any case, the current LEO entry for 'self-taught' is lacking the hyphen. And the above entry is missing.

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