Hi there stone_rose
This nice looking spot with turqoise water is the beach of Pointe d'Esny
Its on the South-East coast of Mauritius at approximately 12 km from the international airport of Plaisance and 6 km from the town of Mahebourg. So you can make your way from Mahebourg easily and just ask people on your way its a damn straight road!
Used to go swimming there. There's a famous guesthouse there:
"Chante au Vent" its called check it out!
http://www.destinationmauritius.com/con ... /sleep.php
I don't think there are private beaches anymore! I might be wrong...hope this helps & ENJOY!