ETOPS requires 2 types of certification. Type and Operator. Type certification comes from the regulator where the aircraft was designed and manufactured. In most cases this is the FAA and EASA. Each aircraft and engine combo needs it's own type certificate.
The operator certification comes from the aviation regulator in the operators home nation.
In LATAMs case for their SCL-AU/NZ flights, that comes from the Directorate General of Civil Aeronautics of Chile.
Qantas would need CASA approval.
A few years ago, AA got in trouble because a 737 planned for a LAX-Hawaii flight, an ETOPS route, had to be pulled for maintenance, and the aircraft they arranged to replace it wasn't one of the Hawaiian subfleet and didn't have the extra equipment required for ETOPS. No one noticed until the aircraft was half way to Hawaii.
Normally new designs are given ETOPS-120 or 180 ratings and increased as the model proves itself (The 777, with GE engines, was given ETOPS-330 rating in Dec 2011, after being given 180 by the FAA and 120 by Europe at design in 1994). The 787 however got approval for ETOPS-330 during testing, but had to make some software changes before they were allowed to use it.
RR equipped 787s have now had their ETOPS rating reduced.
Until CASA approves over ETOPS-180, the southern routes aren't viable with 2 engine aircraft, as shown by the VA MEL-JNB route.