Today’s correspondent was ‘… riding my trike through Bendigo in Central Vic and a NSW Ambulance passed me running lights and sirens. How can they do that if they’re from interstate?’
The answer is that the definition of ‘emergency vehicle’ in Victoria includes ‘an ambulance service created under a law in force in another State or in a Territory of the Commonwealth’ (see Road Safety (Vehicles) Interim Regulations 2020 (Vic) Schedule 2 cl 3 and Road Safety Road Rules 2017 (Vic)). That means that the warning lights don’t have to removed from an ambulance visiting from interstate, and the crew have all the rights and responsibilities of Ambulance Victoria on a Victorian road.
Conclusion
Provided the use of lights and siren was reasonable in the circumstances and they were taking reasonable care, as a vehicle of an ambulance service created by a law in force in NSW (ie the Health Services Act 1997 (NSW)) it was an emergency vehicle and could be operated accordingly.
Australian Emergency Law 
Ambulances respond across borders every day, many also take patients to interstate hospitals on a daily basis. This is business as usual (BAU). Under resourcing and increasing use (5% increase on average) every year means Ambulance services are at almost 100% capacity every day and cross border responses are necessary to meet emergency workload.
That may all be true, but it doesn’t address the legal arrangements in place to ensure that they can do it. For example Victoria has it sorted so that a NSW Ambulance is an ’emergency vehicle’ in Victoria, but they do not have law in place to allow NSW paramedics to carry their drug kit across the border, but the law is not self-executing and no-one is seriously going to worry about it; but getting the law sorted is a good idea – see https://emergencylaw.wordpress.com/2017/09/14/paramedic-practice-in-nsw-and-victoria/