Seriously? I mean, I have no first hand knowledge, but that was not the story / impression I was given by colleagues who practiced abroad before coming here. If a real (read sufficient size -- not supercar status) market exists, how does it not crowd out the government payers?
Private insurance or out of pocket payments for procedures not covered was illegal in Canada, but it got struck down in 2005 by their courts (I think, someone please correct me if I'm wrong).
I've lived but not practiced in England, and some people do carry supplemental insurance. You still have to pay the (very high) taxes that go towards the NHS however, and all in all the government system is relatively comprehensive in what it covers, though the wait times suck. But that's why there aren't tons of people flocking to private insurance. Plus it's fairly expensive, and there's much more of a culture distrust of 'for profit' health companies. The old scare of refusing dialysis to people over 65 hasn't been true for a while.
The NICE does take a lot of heat from people, but at least it's made up of physicians. Legally if you follow the NICE recommendations for treatment,
you can't be sued for malpractice. It won't even go to court.
If you have a specific question about either, or about any other OECD system, France is for example, rated number one, and is essentially a private system...we can certainly discuss further.
I think this may be somewhat misleading, calling it private. I've lived but not practiced there too. Something like 70% of the population does carry/pay for private insurance, but the private sector is heavily controlled by the government. What they can charge and what they must cover is mandated. As a result, many have trouble remaining solvent and in business. I think the government has had to supplement a few of these companies to keep them in business.
One nice thing about it however is that the law requires insurance companies to pay physicians within a matter of days. Prices for services rendered are posted in the office, so you know how much you as a patient will be responsible for up front, and there generally is some sort of co pay.
Of course the French revamp the system every few years.
I'm actually a fan of the Swiss system. Tons of private plans, with some government safeguards to prevent abuse of patients.