Long Bay Beach 2 is pretty secluded, but not entirely private.
It's a public beach and i presume during the weekends, and summer it will be busy, but while i was there ( i just back this past wednesday), i had the beach to myself.

Just North past Cosmos, is Long Bay Beach 1.. then you go up further along a path near the roadway and then come back out onto the beach (LBB2)... It's a nice hike, and perfect destination to cool off at.