I tried it when I first learned chess, and it was fascinating, because I had no ideas about strategy or tactics. But now that I have some experience, it seems pointless, as I know what my "opponent" is thinking all the time, and what my main ideas are for both sides. Should I still make an attempt to play against myself? Is there a deeper point behind doing so that helps one's chess ability? Do strong player (expert or above) do it?
Depending on exactly what you mean by "playing chess against yourself," I would say it can be extremely beneficial, and that yes, strong players do this quite a lot.
Consider what you do when you have to decide what move to make when you're playing a normal game against an opponent distinct from yourself: you analyze the position as best as you can in your head, which means needing to figure out what possibilities are available to both players in order to get to the truth of the position. In essence, you play some portion of a game against yourself starting from the position you're at. But you don't just play straight through; you have to consider sidelines and various ideas and pitfalls that might pop up along the way.
Because this is such a fundamental part of playing the game, analyzing positions or entire games on one's own, freely exploring moves and ideas while trying to determine best play, is a fantastic way to improve at chess. Not only does this endeavor have the simple potential to open your mind by exposing you to more and more types of positions and ideas, but it also offers practice at just the kind of thought process that you continually need to engage in internally during a game. One can anylyze existing games, of course, be they grandmaster games or your own past games, but games initiated on the spot serve just as well to offer up new positions in which you have to figure out how to play for each side.
So if, when you play a game against yourself, you don't merely play an alternating sequence of moves until you get a result, but instead do some stopping, backtracking and general meandering, digging deeply to figure out how and why things could proceed differently, then really you're working to develop your analysis skills, and so it could definitely prove beneficial.
If you remain unbiased, playing against yourself will force you to try to find the best (or at least good) moves for both sides. It is actually a stiff mental exercise.
But it gets you thinking, if I do this, what is the other person going to do, and what do I do now, given that the other person has done this.
The danger is that you will get into a "rut" and learn only certain types of positions that you are particularly suited for.
Yes! I find it extremely helpful to play chess against myself. It is important to remain unbiased. At any given time, I have a chess board on a table which has an active game usually going for a week or two that I play against myself. Playing against yourself will allow you to consider the tactics of your opponent more thoroughly because when you are playing against yourself, your opponent always knows what you are thinking. This can be important to consider, in that, if I have a forced mate against another player in 5 moves, it really does not matter that we both see it, because in the end, the mate is forced. My point is that even if your opponent sees the correct move coming, that does not diminish the move.
Composing chess problems is "playing chess against one's self". Trying to find interesting and novel behaviour in the arrangements of pieces. It can be exciting and frustrating, and the feeling when the plan finally comes together into a real position is incredible. The competitive aspect, if one wants it, is that other composers are also searching for cool stuff.