Actually there is a public toilet and shower-room on the beach and has been for years! Not many farangs know about it though, even if they pass it buy everyday!
There are often foodvendors in front obstructing the view.
It's on the middle of the beach next to the beach-restaurants opposite the Soi Sawatdirak crossing. It's 5 baht for the toilets and 10 baht for the shower. As with everything in Thailand it was built for "pygmies", so the shower-tap is only about 1.50 meter above the floor so you have to sit down to take a shower!
It was destroyed by the Tsunami but was repaired again when I was there in April. Some years back there used to be open showers all along the walkways behind the beach, where you would put 5 baht in an automatic coinoperated shower allowing you to shower for a few minutes! It didn't last long though until people had kicked doown all the coinboxes to get the money!
I agree that one place is not enough should be at least 4-5 shower-rooms and toilets along a beach Patong's size!