Free camping has traditionally been the way that backpackers stay on the islands - that, pay-by-person+tent camping grounds, or sleeping on the roofs of hotels for a reduced fee.
Years ago, I camped on the beach on the Western Naxos coast, which is run by the same municipality as the islands youre interested in, but these municipalities have become rather fascist in their intolerance. But to my knowledge it was never a national law that had the final say (even though it is a law, and ridiculously, that applies to homeless people as well).
You may well find that off the beaten track, say a few coves along from where the main beach roads dump visitors to the sand, nobody really cares. Theyre often more concerned that people don't gather in a collective camping zone, for various bullshit reasons but also due to leaving trash. The problem with Hellenic islands is that they WERE a great place to visit, before generation X screwed everything up. This country is the spiritual home of what used to be good and is now totally gone (usually due to Westernization, EU and its supporters, Karamanlis family, etc) due to betrayal. So the difficulty with sleeping on beaches is that most of them have a **ing beach bar, which means the bar thinks they own the place, which means they get territorial.
HOWEVER: A few coves over, there may well be a taverna which, if you use them as your regular food/drink provider, they might let you use their facilities. I know someone who kept his valuables with them, so striking up a relationship is probably the best thing. Location matters - most of the successful campers Ive seen were away from the trash stretch (and for some reason, usually the best bet is near or on the nudist beach).
Fee-charging campsites these days charge a ridiculous amount of money - as much as youd expect from a hostel in some cases - and you should contact them ahead of time to see if bringing your own tent will cut out the per-tent fee. For what its worth, most official campsites are listed on www.greececamping.gr - although stupidly, this encourages that disease of booking ahead as well.
My friend and former co-party organizer makes a beach teepee on one of the islands every year, (hes also Greek)and his relationship to the local business owners plays a part. They have campfires, guitars and full-moon parties with foreign tourists, everyone has a great time :-)
Good luck, tell me how it goes eh?
D.S. Dando