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.)