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You can go to Platia Ammos beach which is here (photos are geotagged):
https://picasaweb.google.com/103408914622400708805/TinosPlatiaAmmosBeach (Sorry! this no longer works…. )
Drive (or take the bus) to Kionia. Continue driving (or walk) after the main beach. The road starts to go uphill. You will see the beach from a certain point. From that point on, you have to continue by walking down the (not so difficult) pathway.
The beach has a southern orientation which means it should be protected from the strong northern winds blowing in the Aegean in July/August. It is a very beautiful beach.
2018 July: really nice wild beach - no facilities whatsoever, and not much, if any shade - I didn't see anyone here nude, but then I saw very few people at all here during a week in July 2016, and again in July 2018 - lots of room to be be nude without the crush of others.
2024 June: Two nude couples and some clothed dog owners on the beach, which is rocky and not really appealing.
Editor
September 2024 – Barefooter submission: nice beach, but narrow, due to some days with wind and high waves, which reduced the beach to a narrow stretch. Maybe the sand will come back by winter.
Be aware that there will be big waves each time a ferry passes by!
Parking your car is difficult at the path to the beach, as the road is very narrow here, but just before the next left turn in the road (coming from Kionia), there is a small stretch left of the road for 2-3 cars.