Should playing chess against yourself always end up in a draw? You play against yourself, using your own opening repertoire.  Thus by playing both White and Black, you reach positions that should be fairly equal, and then you continue to play both sides, obviously knowing each sides' plans.  Should the game always end in a draw?  How often is White expected to win?
No.

When you are playing seriously against yourself, you still have to commit to moves. After making the move you start analysing again and see things you previously have not seen, and you may discover an error in the moves you have already made and the game becomes a win for one of the sides you play. 

Or you may find a winning way without really catching an error in the previous moves.

Note that there are no secret plans in chess, the full truth is always on the board. It is up to you to read the board and draw the best consequences of that reading.