Good question. May I use a personal example. I had causae to see my GP once, I knew what it was. He confirmed me (I had to pay his the balance of his fee and the money that the Medicare care scheme paid, then about $20 difference)., I had to see a surgeon. The surgeon had a look and told me what I knew. He then asked me if I had private insurance. I said what's the diff? He said I get to go into a hospital of my choice, as soon as he can fit me in and he does the operation. If I went as a so-called "public" patient then I would have to wait several months (it wasn't life-threatening or it would have been done immediately) but he would do it at a specific hospital. I had insurance so it was done that week. I didn't see a bill. Between Medicare and my private health insurance I didn't pay a dollar.
To sum up:
A. Non-life threatening, a patient without insurance will have to wait, then go to a hospital they are told to go to but the surgeon they consulted will do the op. Cost zero.
B. Life-threatening, a patient without insurance will get immediate attention at the nearest hospital, will get surgery or whatever they need. Cost zero.
Private patient in A - gets surgery quicker, hospital of choice (get frills like cable tv if you want it), same deal with the surgeon though.
Private patient in B - no difference. Get treatment, no cost, it's an emergency.
You can get dental treatment for zero if you are willing to be a public patient at a hospital and wait. If you have insurance you go to your dentist and get it fixed.
It's a little more complex than that but in very broad terms that's how it works.