Question: I've started to teach myself how to play the piano, and one of the difficulties I've found is trying to get both hands to do things at once. I either forget one hand entirely, or have one hand trying to mimic the other. 

This reminded me of many years ago when I was taught to play the viola. For many weeks, we did not use the bow, instead focusing on getting the left hand to do the things it was supposed to, and learning to read music. In hindsight, this makes sense to me, as it's often important to get used to some fundamentals before attempting to do everything at once. 

So I was wondering, when learning the piano, is it better to learn each hand on their own, or muddle through the difficulties of synchronizing them from the beginning? Or, if there are arguments for both methods, what are the pros/cons of each? What problems may develop, or what difficulties might be avoided? Or is there perhaps a happy medium that has proven the most successful?

For my personal musical background, I have about nine years' experience with the viola, though I haven't played it in about five years (stopped after high school). I also play the guitar, but never learned to read music for it.

Answer: 
The mantra in learning piano is: "start slowly." How slow is slow? So slow you are unable to make mistakes, and then a little slower than that. 
Invariably when I have to choose a speed at which I think I can't make a mistake, at first I think: "come on, this is ridiculous!, of course I can't make a mistake"... and then I make a mistake.

As for the specific problem of hand coördination:
In the very beginning it helped me to practice away from the piano at first with my hands on my knees. For every note in my left hand I slapped my hand on my left knee and vice versa. When I could do that, I played the right hand on the piano and simultaneously moved my left hand in the air (or slapped it on my knee). After that I switched roles. The next step was playing with both hands at the same time, slowly.

After several weeks this kind of exercise wasn't needed anymore, but in the beginning I found it helpful to 'rewire my brain' so to speak.

Concerning hands separate versus hands together:

For any piece 'at your level' beginning with practising hands separate is needed. It's being used to tackle problems of reading, fingering, dynamics,..., before adding the extra complexity of having two hands to control.

But what I experience is that playing hands together after having practiced hands separate can feel as starting completely over. What I mean is that passages that were comfortable hands separate, are feeling clumsy again when starting hands together. Sometimes it is like having to learn the piece all over again. So the need to start slowly (and even slower than that) is there again but now hands together. 
Because of that in my experience it is not efficient to practice towards 'perfection' hands separate before beginning hands together.

Of course the process of practicing hands together is greatly shortened (or even possible), because of the knowledge you gained from the practice hands separate, and that's what I use it for: orientation, experimentation and planning. But really learning the piece only starts with hands together.
I cannot gauge your piano level very well, but if you are having difficulties playing both hands at the same time, then the piece is may be too difficult for your level. 

For piano players that are beyond the elementary level, I believe most players usually start learning the song both hands at the same time. This does help you get to learn the timing of the right and left hand right from the get go, instead of having to piece them together later on. Learning each separately and then piecing together later may actually be slower (depends on the piece) because of complex interweaving melodies, etc... 

However, the practice is often done separately as needed. My piano teacher used to say if you cannot play the one hand alone, you definitely cannot play it properly with two hands. When practicing one-handedly, it will often reveal mistakes that you don't really notice while playing with two hands.

For elementary players, (at least from my experience), usually the learning process is just:

* right handed songs;
* primarily right handed songs with some added notes on the bass clef
* more difficult songs with a more involved bass line for the left hand

That's why as I said earlier, perhaps if you are having difficulties of left and right handed songs, then maybe you are playing songs with a bass line that is too difficult at the moment for you.

One thing you can do to improve independent motor control with your hands is practice your left and right brain. The left brain controls the right hand and right brain for left hand. One exercise I was taught is to try rubbing your right hand on your thigh in a forward and backward motion (relative to a sitting position), and with your left hand do a upward and downward motion in a fist. Switch it up to have right hand as a fist, and left hand doing the rubbing. I know it sounds and looks retarded, but some people actually have difficulty because it involves independent directional motion.