I don't believe that you can technically teach yourself to sing. Either you can or you can't, either you have it or you don't. I do believe that you can fine tune your singing ability with proper direction, once it is established that you can carry a tune for more than just a few notes.
If you sing a bit pitchy here and there, it can be corrected. If you sing totally off key and you're all over the place, you probably need to find another hobby!
Before my voice broke, I sang in the school choir. Then I discovered The Beatles (after they'd split up, of course, and I was too young to remember them in their heyday), and I was DYING for my voice to break so that I could sing like them.
What happened, of course, was that my voice broke, and it took about 4 years for me to gain any kind of control over it. Even though I played guitar, I sounded like a yodeler hitting all the wrong notes. My friends used to play demo tapes of me for a laugh when I wasn't there.
However, over the course of the last twenty five years or so, I have not only learned to sing mostly in key, but - and I'm quite proud of this - when singing cover versions/other people's songs, I sound like me, not like me singing a Beatles song.
I'm in no way a natural singer, and I need an instrument to keep me in tune/on pitch most of the time, but I have learned to (1) technically control my voice to a large extent, and (2) use my voice to develop a 'voice' or way of singing that is individual (I hope I eschew falling into the trap of establishing a style).
Now, I'm applying the lessons I've learned to try to develop my ability to sing harmonies - something I had previously believed was impossible to develop. It's going slowly, and is easier in terms of hearing and singing harmonies that come naturally (fifths, I find very easy), but nevertheles I'm improving.
So, Trumpspade, based on my experience of being off-pitch/key 95-99% of the time, to being in tune/on pitch about 85-90% of the time, I'd disagree that one can't teach oneself to sing.
Now, I've seen my cousin - who's a much better natural singer than I am - be helped tremendously by lessons, too, but I'd say he had 'it' to begin with (by which I mean a kind of arresting quality to the way he sings, regardless of whether he is on pitch or not).