I think you usually say "I'm self-taught".

"Self-taught" would be used for a specific skill. It is a common word. It is probably the word you want. "Self-educated" would be used for an entire program of education. For example, someone who had never gone to school at all, but had learned all the elements of a full education through self study. It is a rarer word, because self-educated people are rare. A self-educated person is an "autodidact." For example, I am not self-educated, because I've been to (many years of) school, but am self-taught in singing, because I haven't had formal singing lessons. The famous US President Abraham Lincoln "was a self-educated lawyer with less than a year of formal schooling." Victor Buso, who discovered a supernova, was "a self-taught astronomer," because although he had been taught other things, he had learned about astronomy entirely by himself.

I think that "education" relates to all education post-secondary and ends with a diploma. In that sense, I can't imagine what "self-education" might be. Self-taught usually relates to skills: mechanics, languages, nutrition... If you're looking for a word that says that a person has learned alone, I would say "self-taught"

Different variants with the same meaning, you can say: - I'm self-taught - I'm self-educated (correct but a little formal and infrequent) - I learned by myself - I learned on my own - I taught myself - I'm an autodidact (say this if you wanna show off)