How about all of the above?
I started as a musician, playing professionally beginning in high school. Being interested in the technical and having been a young tinkerer with basic electronics, radio, and audio, I paid special attention to both the live sound crews I came in contact with and ultimately the engineers in the various studios when we started recording. Later, while attending music school, I decided to major in audio. Also around that time, I and a friend started an 8 track analog studio where I satisfied a lot of the "self taught" part of the equation. After that I interned in an SSL studio and later in a mastering studio. I'd have to say that "all of the above" best describes my foundation, and I think the well-rounded picture one gets from such varied experience early in one's career is quite beneficial.
There seems to be a big debate about school vs. internship (and I don't mean just the month or two they make you do at the end of a many school programs). However, I believe neither alone is truly sufficient given the modern dynamics of the biz. You should have both these days to really get the complete picture and to be best prepared for the highly competitive and quickly changing marketplace. My advice to young, aspiring engineers is to get a college degree studying relevant subjects (music, audio, electronics etc.), and then spend some real time working in a serious studio, applying yourself, soaking up knowledge, both technical and business (including client relations), and networking. This is good advice for not only a career in audio, but life in general.