Folks,
Since about 1982 or so, ALL aircraft registered in Australia have to have a Noise Certificate or a Noise Exemption Certificate. See the Air Navigation Act.
Aircraft on the register prior to 1982, that do not meet Noise Certification Standards (and there are many, almost anything with an IGSO-480 or IGSO-540, just for a start) are exempt from meeting the standard.
All Experimental, Restricted or Limited Cat. aircraft require a minimum of a Noise Exemption Certificate. If it is a CASR 21.189 (Limited Cat C.of A) aircraft used for "Adventure Flights", additional rules and restrictions apply.
Tootle pip!!
PS: "Private" (a category of operation, not registration) makes no difference.