I live in Western Australia and it's all done by St John's Ambulance. My partner has needed two non-urgent ambulance call-outs (transfer to a public hospital), which were both charged at ~$500. She's on my private 'extras'* insurance policy, and my insurer fully covers an unlimited amount of ambulance transfers ($50 co-pay for non-urgent). I believe this to be a common thing for this type of insurance policy - I'm not on any fancy tier of coverage.
HOWEVER:
A couple of years ago my significant other had an accident in a public area where she unable to move or call an ambulance. A nearby resident heard her yelling and called the police / ambulance. The paramedics seemingly had the authority to take her away from the police officers who were questioning her. She was taken to the nearest public hospital and was discharged from the ED shortly thereafter. She never received a bill. She's only young and the accident (very apparently) occurred as a result of drug use, so I think that the paramedics operating the ambulance are allowed discretion when deciding whether or not somebody should be charged.
* Physio, psychology, dental.etc. (all non-emergency). These services are not (fully) covered by our public health system, so many people opt for private cover. These policies work heavily on annual limits, so it's a bit unfair to call them insurance.
