My wife and I (late 20s) are headed to Cozumel as part of a RCCL cruise in July. We are debating whether to take a beach shore excursion (a boat rental to hideaway beach)or just rent a jeep and head on our own. Its our anniversary and we are without kids so we are looking for some fun private time. Any private beaches to explore? Things to do? Thanks.

Most definitely rent a jeep and tour the island.
Anything you do with the cruise line will be jam packed with people.
Driving is easy because the island is small and there's really only one main road that goes around the island.
The best secluded beaches are on the other side opposite the town of San Miguel.
In July, you should have no problems finding a stretch of beach all to yourself, but don't go swimming on the other side - the water is very dangerous.
If you want some facilities (chairs, palapas, bar, etc.) you can find some nice places along the road on the San Miguel side up from the Presidente hotel.
There's Mr. Sancho's, Palancar, Playa Sol, Playa San Francisco, etc.
Find out where the people from the ship are going and go elsewhere.

Do avoid the overpriced and crowded ship excursions. They will all stay on the west side of the island. Rent a jeep or car and head for the opposite side--thats where the beautiful , secluded beaches are located. I would suggest you ask someone how to find Chen Rio--a beautiful beach and great seafood and drinks in a rustic setting.

Enjoy!

I believe we will be renting a jeep. However, in looking at maps of cozumel, there seem to be two dirt roads, one southwest from playa bush on the south end and another that heads northwest towards the punta molas lighthouse. How will traffic be on these dirt road. Will there be lots of traffic heading on jeep through this area. Is there enough seclusion/playes to pull over or just a madhouse of jeeps heading through there?

The companies that rent the jeeps tell you NOT to use the dirt road, and they will charge you a hefty fee if they find out you did.
I talked to a couple there last year who had friends that decided to go on that road anyway and broke the vehicle's axle.
They could not leave the island until they paid for the repairs in CASH! ($3000+)
There is so much more to see anyway on the paved road which covers more than half of the island and then cuts across the middle.
If you are interested in seeing the other portion where the dirt road cuts through, take an organized ATV or Jeep tour where they know the road and what's back there.
I've been on that and can tell you it's dusty, takes forever through jungle and rutted roads, and the other side is much nicer with deserted beaches and little bars.

Good advice from Diana! You will find plenty to see in the way of beautiful beaches without taking the dirt roads. This is very risky.
The paved road on the east side of the island is not crowded--most choose to stay with the masses of people on the west side!

Great to hear about the east side. Just want to get away from the crowds. I remember going to Orient Beach in St Marten to relax a few years back and the buses with the cruise ships just dropping people off to watch us enjoying the beach.