Wait for their hands to stop moving.
that would be when they dip under and don't come back up? They're lungs would be full of water then? So you would have to swim down to them, pull them to dry land (they won't float) and resuscitate them?
They won't sink at this point, they'll just float with their face in the water. (There is a reason why the dead man's float is called that. See http://www.ehow.com/how_6582_survival-float.html for a description.)
And yes, at this point they'll have water in their lungs, and may need CPR.
I understand sometimes that could be the only option... but seems like a heavy responsibility.
If you're not trained, then getting in over your head (literally!) is an easy way to commit suicide. I had multiple years of lessons before any instructor gave me any instruction on how to go after someone in the water beyond, "Don't."
Now one thing that I didn't talk about was how you can tow someone in the water. If you have any swimming training you may be familiar with the elementary backstroke. (If not, hopefully you know the breaststroke, basically it is a version of that except on your back.) If you continuously do the kick from that, you get the whipkick. (Well to get it right you need to keep the knees a little closer together than many people's breaststroke, but close enough.) It turns out that the kick is enough.
Now what you do is lay the person with their head on your chest, your arms coming under their shoulders and crossing their chest, and just do that kick as fast as you can. You won't move very fast, but you can keep both your face and theirs in the air as you go. If you get them early enough, there is a good chance they will start breathing on their own as you tow them. At which point you get them to land and your job is done. If not, you'll get to land alive without having done further damage and can start CPR. (A life skill that I firmly believe everyone should have.)
One of the ways I have fun in the water is to breathe a good amount of my lungs out, and sit on the bottom as long as I can. Sit, not hold-by-swimming. It's an incredibly effective way of learning how to control your heartrate & panic response to holding your breath, because it's hardly possible without control, and then it's pretty darned easy.
"Ommmblblblubublbubublublb"
When in a dangerous situation, your first responsibility is to yourself. Rescuing someone else can be risky, and getting yourself into the same situation as the person in trouble helps no one, and increases the burden on any other would-be rescuers, whether professional (lifeguard, fire fighter, etc.) or another Good Samaritan like yourself.
But I agree that it seems like waiting for the drowning person to become unconscious dramatically increases the chance that they won't survive. That's gotta be a tough decision to make.
As such, I would say it's not unreasonable to wait for the drowning person to lose consciousness if you aren't confident you could prevent or deal with him or her dragging you down.
However, the good news is that drownings are one of the few times where Hollywood-style CPR "Do a couple rounds and they wake up" can occur. Their heart stopped beating because it stopped getting oxygen, not due to some other type of damage. So if you do CPR for a little while and get their heart oxygenated again, it can start back up in its own (most pools nowadays have an AED as well)