The entirety of California beachfront, from Oregon to Tijuana, is public property and you are free to hike its entire length, swim in it, stick your feet in the surf, make sand castles, have picnics, fly kites, go crazy. However, there are places where you must pay to park; or where you may not smoke (in L.A. City, L.A. County, Malibu and Santa Monica). There are also restrictions on dogs, fires, and sleeping (not a good idea day or night). Also, while the beach is public property above the mean high tide line, David Geffen has mean security guards who will hassle you if you walk past his house. Bring your sunglasses, hat, swimsuit, towel, and rented convertible, as well as drinks, snacks, and plenty of SPF 30 sunscreen.

Leo Carillo State Beach, about 15-20 mi N of Point Dume State Beach, has beach camping (across PCH). All you need is a sleeping bag.

Beaches are not mountains, nor are mountains beaches. If I were you, I would visit both. Set your GPS to the intersection of Stunt Road and Saddle Peak Road.

Please do explore Point Dume State Beach. Look at the satellite map. You will have to pay to park near the point itself.