"[...]how to get someone who wasn't quite drowned, but definitely choking on water, to breathe properly, right?"

Assuming the victim has been through a near-drowning experience and has inhaled a significant quantity of water while in distress, the proper response would be to get them to a hospital ASAP. You're not trained to go and drain somebody's lungs, you're there to make sure the paramedics get someone who's still breathing.

You should call for an ambulance, describe the victim, give your address, ask for ETA, and send someone outside to guide the paramedics to the victim. Get someone to fetch you a blanket and a first-aid kit, and put on some gloves before touching the victim again. While waiting for the ambulance you should treat your vic for shock, maintain and monitor vital signs, and inform him/her that they can suffer from secondary drowning, where inhaled water can actually cause you to drown on dry land up to 72 hours after nearly drowning.

I still use my trusty Canadian Lifesaving manual for reference. If you live stateside, I'm sure you can either become a guard yourself (it's really fun!) or find their textbooks somewhere at a pool that gives lifeguard training.