Take the bus from the pier into town. Get off the bus. Head east along the Malecon so that the water is on your left side. Keeping looking at the water. When you find a spot of water you like, turn left and walk toward it. Lay out a blanket. Apply sunscreen. Repeat as necessary.
I don't know of any specifically free beaches on Cozumel except for the lightly visited east side, but there's a few with low cost entrance fees and pay-as-you-go extras. Paradise Beach, maybe? It'll take a cab ride to get there. But it'll be considerably less than $80/ea.
SIG: If you're going to create the drama, at least supply the popcorn.