I've been here a couple of times, most recently in June this year. Catching the free boat around the coast to this private beach you are treated to some amazing views. I've not had a problem with getting a beach chair but perhaps I'm going earlier than others. The beach chairs (lettini) are EUR7 which I don't think is unreasonable, most places will charge between that and EUR12. The shore is all pebbles which is pretty much the case everywhere in Italy other than Calabria and Sicily so if you're used to sandy beaches then be sure to have your thongs/flip-flops/jandles with you. You can tell who the non-locals are as they're the ones like me keeping their shoes on right up to the water then falling over when the water is ankle deep because of the instability you feel on pebbles :-)
Lunch is definitely the highlight, the menu is on a blackboard & all in Italian, we had to help translate for some other tourists as they weren't able to catch any of the wait staff who were running around trying to keep up with orders. I would recommend doing your research first or having a phrasebook/dictionary with you. All the waiters I've dealt with speak English but it'll make your menu decision easier.
I usually have cozze (mussles) for entrée, they do them multiple ways, I have the zuppa which is in Napoli/tomato sauce. Main is always spaghetti con vongole (clams), is basically just oil & garlic for the sauce and it's delicious!
I believe they are fishermen who cook what they catch that morning, the seafood has always been beautifully fresh.
The Italian way is to sit all day on the lettini and bake, hence the first boat back only being at 4pm, unfortunately I married a guy with Scottish background and they don't go well with sun so we got a water taxi back which was around EUR15.