It is IMPOSIBLE to self-initiate. Fuss all you like, disbelieve, rail, cry "pagan persecution" and "meanie poohead traditionalist" until you go blue in the face. It is impossible to self-initiate.
Initiation requires that there be a trigger and catalyst. A indoctrination by one party, of another party. A singular individual cannot indoctrinate them self into what they do not know, and if they already know it (someone self-initiating into their "own tradition") there is no induction process. One can dedicate them-self toward a practice, or confirm their walking of an already-established mode. But that is not an initiation.
But, of course, it's possible to be:
Initiated without a -physical- Initiator.
Initiation is something you can do by yourself, but it is something that can occur when no on else is physically near you. All circles are as one circle, and if properly cut through the worlds into a shared space, two people half a world apart can join together in spirit to pass the proper ties. Some traditions allow for this, though all the records are kept, and when the time comes the person ideally undergoes the physical initiation as well.
In addition, The Gods may come when sincerely called, and jam the right wires into your back. And though that may sound a little "harsh", direct god-to-human initiation is not exactly gentle. It's big, it's rough and painful... and often far more filled with life-upheaval than even the most misguided of human-on-human initiations. Why?? Probably because the experience of a body is not their strong suit, and thus they are unaware of the finer points of keeping it in perfect shape (or, in the case of some gods, the point is breaking you apart so they can remake you). Maybe because they are -gods-, and the small sip of energy we are accustomed to dealing with is nothing compared to the torrent they pour into us.
These initiations are really not ideal where most Trads are concerned. They are messy and forge connections which my folk call "Nonstandard in either quality, quantity or topography" (as compared to a "normal" initiation). The understanding of the Gods/Powers, rites, rituals and purposes of each, which come through these connections are most often NOT in holding with the Tradition one was trying to jack into. Hell, even the best-laid initiations sometimes result in this.
Self-Dedication, versus "Self-Initiation"When I was a very young witch, I did a self-dedication ritual from a book. This was not an external experience (like initiation), but an internal one. This was a self-affirmation of my dedication (or, gosh, self-dedication). This got me walking in the right direction, and put me on the radars of the right powers (who put me on the radars of the right Priests).
Some folks say that The Witch Power is always inside us, and it merely takes a transformative experience, and applied use of said power, to create a Witch. That's fine and dandy, but still does not cover the grammatical and logic errors of "Self-Initiation". Revelatory experience is a much better term for that.