Trachia beach
Trachia beach is essentially a strip of land connecting the homonymous peninsula with the rest of Halki. It actually consists of two beaches, one in each side of the strip, washed by 2 seas. One beach is with lots of debris washed up, and the other is a pebble / rock beach. Both beaches are isolated and swimming here is just great. There is limited shade but there are a few rock covers, like little caves. Clearly there are no facilities here, so take water and what you need. You can reach the beach from the port on a water taxi or a caique. If you are looking for a totally secluded beach to be on your own, then Trachia beach is definitely the beach you are looking for.
Coastguard: (22460) 45220