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.