I don't know why everyone is making such a fuss about private healthcare provision. I had it as a student (way back in 1979) when doing my year abroad as part of an MFL degree. I think it cost something like GBP 125 for a year.
I currently live in an EU country where healthcare is "private" - except that it isn't really.
Those, who are employed, pay a % of their monthly earnings to a designated agency (often associated with their area of work) or to their employer's healthcare scheme. The employer matches this contribution. If you earn under a certain amount per year, that is it. You can take out additional cover yourself, but most people do not.
If you earn over a certain sum, the above still applies BUT you get to choose the best cover for your circumstances from amongst the private healthcare agencies. Premiums vary according to your "risk", the number of claims, your excess etc. To give a concrete example, I have an annual excess of €1200, but an annual premium rebate of around €1000. Consequently I generally just pay the bill myself (very rarely go to the doctor's, but currently having some dental work done) as it wouldn't be more than €1200 over a year and is mostly covered by the rebate anyway (or I make a bit of a profit).
Medical bills that you pay yourself and for which you are not reimbursed are also tax deductible.
I could shop around for the best deal - currently have a really good one at just over €510 per month - for those of you astounded at the cost, a colleague of mine, who is a couple of years younger than I am, pays nearly double.
The only real difference is you pay your monthly contributions directly to an admin agency.
There are national, regulated scales of charges for work/treatment, that doctors and dentists apply to their billing - all of which are transparent. In addition, you generally get an itemised treatment and cost plan in advance.
But on a cautionary note, there is no bottomless pit of money. My best friend was working in Switzerland and was diagnosed with cancer. The maximum amount his insurance would have paid out on treatment was CHF 2 million. Sadly his cancer was not reversible and he died.