It sounds strange and messes with your mind like you said, I played against myself before. How long did your game last before one white or black won? We just shared one of the best kept secrets because after learning how to play chess...
Learning how to play chess is easy (in a sense). Learning how to play chess against one's self is not, because every time you change sides you have to think about what the other side is going to do. This is absolutely counter to what chess is all about, because when YOU are the other side, you have to constantly counter your own thinking, in order to beat yourself at the game.
To answer your question abut how long it took and which side won, I do not recall. My last foray into that mind experiment was some years ago, and it took me so aback into how fucked up it was that I have not gone back to it since. It really is a mind warp to any serious chess player, or serious analyst of any type. But it does at least give a poke into one's own psyche.
If I have time left in this life, I will try it again. But for now I am trading. But I think anyone who plays chess should try it at least once, and be HONEST with their moves. Take the time. Think though the game, think of the board against your opponent (which is yourself), and counter what you would do against yourself.
It's just bloody hard, and harsh. But hey, one may learn something about themselves, and that is important!