When getting a landline with Telstra, and I think most providers in Australia, you'll be opted in to an entry in the Sensis whitepages (both print and online). You're asked if you'd like to opt out but it costs A$2.93 a month for Silent Line. That'll get your entry removed along with CLID not showing up when calling out.
Following on from this CrowdSupport post there was mention of asking for an UNLISTED line. Conveniently all the representatives I spoke with said that wasn't available and you'd need to fork over for Silent Line. That felt a little like extortion to me...
After a few more tries I have it on good authority that you need to request No Entry Line for Whitel pages Listing. That will be available at no charge and your number won't appear anywhere.
The consultant placing my initial order wasn't aware of this so as soon as my order completes (as it can't be done while it's processing) I'll be phoning up to get this added and Silent Line removed (and refunded).
The consultants seem to be well trained at pushing this service so at times it can feel like there's no option. Persist and you should be rewarded.