I'm in the UK where the climate is quite different from the Australian one, so I can't recommend a plant from personal experience. However, a quick online search has come up with the following plant, which is suitable for Southern (and presumably Western) Australia and seems to meet all your criteria; it is fast-growing and drought tolerant, requires very little maintenance and would provide the privacy you want:
Apparently there are quite a few varieties, so you could choose one that suits your particular situation; Bambusa oldhamii, for instance, is an evergreen variety that grows up to 20 ft.
Clumping bamboo, unlike running bamboo, has a non-invasive root structure, so there is no need to create a physical barrier or prune its roots to prevent them from spreading into your neighbor's garden.
There are two very useful articles about bamboo care and maintenance here and here.