Cozumel is not exactly a beach place in the touristy areas. You can pay some ridiculous fee (ie $25/person) and go to places like naichi cocom, palancar beach, paradise reef etc etc. All those beaches offer either free admission but you will be very much persuaded to get drunk to the boot with high cost drinks or you pay an entry fee and then they make you pay for all other things too.
On the west coast majority of beaches are pay to use beaches that belong to resorts, hotels, bars etc.
If you really want to experience picture perfect beach let me advise you a few sports.
rent a car and drive to Punta sur. There is really not much of stuff to see along the way unless you stick to the coast on west side of island. You can take a quicker way and just take rafael el melgar all the way to punta sur.
Between amazing view, amazing beach and bar and goodies shop that place is worth its weight in gold. I have been there about 5-6 times now at different times of day. Meeting sunset there is amazing. During good weather that place is even better.
From Punta Sur (which is free) you can drive up (be sure to take the right road next to the beach, not the high way) you have about ohh 25 miles or so worth of absolutely spectacular windward shore line. When I say spectacular, I mean picture worthy. Nothing but one giant shore line filled with white sand that makes hair on the back of your neck stand up. The downside is that there is absolutely no shade... like none. As you drive up north there are a few bars and even a hotel (only hotel on that coast). Chen Rio Beach has been recently cleaned up and 2nd bar was built and that is about the only place there that has any shade. Sit under the palm trees or under cabanas, enjoy awesome beaches and scenery.
If you feel adventurous you can follow the shore line all the way up north (on a 4x4 car) I have never been there but it might be worth the drive if you are there anyways.
If you ask me, I would rather pay $55 for a car rental + $10 in gas and go where legs carry you instead of taking a $12 taxi, paying $25 entry fee then have like 9 bazillion people there.
Talk to someone who has been up that north road recently before you try. I think it's more of a washed out trail, and your rental insurance won't covered you when you wreck up there. The army probably still blocks people from trying anyway.
