Healthcare in Australia

Yes is very good

Private health care gets you a quicker heart operation, other wise there's ten month wait . You could go home to your rural township and get it quicker.  on a shorter list.
Always have private insurance, because  the National medicare availble to all does not cover everything..
I admitted my mother for a heart operation in St Vincent's private hospital, that was the only way to get  her op up the list, but still a 3 month wait, we were lucky to get the late  Dr Victor Chang  to operate, his fee was only - modest$15,000. The rest gobbled up by the hospital.
I over heard as I coughed up $ 75,000 , a conversation next to me. The unfortunate woman was told her medicare didn't cover cost incurred for iCU, post op single room stay, meals, medication and miscellany. The client was surprised by the fact she had to find another $45,000 she didn't have.
I fly myself off to Bumrungrad a Hospital Bangkok for all surgery, plastic and other upper thoracic surgery now,ndental treatment and other needs. Women die in hospital admittance lobby toilets waiting to give birth. My mate had to get a reverse mortgage to fly to Germany for treatment, after 15 years of botched surgeries and diagnosis of what was as simple as Lyme disease variety bepsiosis, not recognized by  Politican ( Tanya Plebisek) I had personally written to, as sufficient evidence to support the link between the condition and Lyme disease.
Overall, don't hold your breath about general medical services, I ve many friends who can't afford dental work badly needed. I am flying to Bangkok to get  bridgework done at half the cost at top rated dental surgeries.
In short Medical services suck a big one in Australia. Have spare airline tickets in your drawer, ready to fly out for medical treatment.
God forbid that you ever need Australian Medicare cover!

By the way. I had no recourse to insurance nor Australian, medicare cover for my mothers operation, she wasn't a citizen, it's all private wealth that buffers us from the cruel realities of life.

GPs are generally very poor at diagnosis - if you find a good one, they're like hen's teeth. They will jump to conclusions based mainly on age. Do your own diagnosis on line before going and question thoroughly.
Otherwise work is very expensive - two to three times comparing to Europe. Only issue is the cost of travel/accommodation probably not worth it, so they have a captive audience. Thailand is very good, super efficient and reasonable cost from my experience with minor surgery. Best deal would be expat medical systems - buy before you arrive.

1.  how does the healthcare system work in Australia ?
It's similar to other countries.  Your first port of call is a GP and then he/she might
recommend you see a specialist, if he can't fix it.   If you have Medicare card the
consultation may be either free or subject to gap payment.

2.  Is it efficient ?
What do you mean by efficient? I've had good and not so good experiences with diagnosis.  It was an Australian doctor in Sydney who identified my sleep apnoea, after doctors in UK and USA, as well as Melbourne had failed.
Australia was the centre for sleep disorder research.   

3.  What are the main differences between public and private sectors?
Waiting times for hospital treatment.  Specialists are private but medicare will cover
most costs.  There are public hospitals and private hospitals.   Waiting time is shorter if you pay.

4. Is it recommended to purchase private health insurance in Australia?
It's a personal choice but Private health insurance is complicated.  I had it for a few years.  It
was expensive ahd I would have been better off paying cash for treatment than using insurance, because
there is always a gap or co-payment.

As a UK citizen aged 64, on a 457 visa, I consider the Healthcare system to be good. We only have private healthcare for emergency ambulance and dental. Husband recently had a quadruple bypass fully paid for by Medicare as part of the reciprocal agreement with the UK. I've had annual breast screening because of family history and recently had a couple of endoscopies, and various x rays for arthritis... All free with Medicare. We couldn't afford the $2000 a month quoted for private healthcare but are well satisfied with GP and hospital services. The GP bulk bills as have the other services used, so no cash outlay ever!
Dental services are expensive but we are very pleased with the dentist we have and consider the quality of the work better than we're used to in the UK. We feel HBF gives us good value for money with the private cover we've got.

Eyecare is also vastly superior to UK at roughly the same price for better quality.