There maybe a couple of strips we're talking about here. The one with the sealed runway, hangar at the western end, and very nice looking home overlooking the ocean is definitely privately owned.
The other one on the coast is to the south of Bermagui around the Murrah lagoon inet. I did a charter there in a C310 nearly 20 years ago. It's low lying and runs N/W-S/E. about 600-700 metres (from memory). It's a grass strip, and although I had concerns about the surface, I found it firm and well drained. There are power lines perpendicular to the approach from the East, and significant rising terrain to the west! The operator I worked for at the time gave me a comprehensive brief, and frankly I had my concerns. In the end the old 310 handled it well.
Can't tell you who owns it now I'm afraid.