Our healthcare system has some similarities to both Canada and the UK, and some similarities to yours in the US.
We have “socialized” public healthcare, funded and operated by the state (read: taxpayers). It is excellent, but there are waiting lists if you are chronic, which are triaged and prioritized. If you are acute you get seen right away.
We also have private healthcare, where you can either pay yourself or more typically have your health insurer pay. It too is excellent, and there is never a wait.
I have used both systems in New Zealand: it is quite common for Kiwis to have health insurance. The public system, as often as not, uses the same surgeons and nurses as the private system, the facilities are quite good and the food is superb (I often ask for seconds or thirds when I’m in).
The private hospitals are like five-star hotels: they even serve wine with your meal.
Socialized medicine isn’t the same thing as Socialism: it’s actually a very good thing and quite efficient if it is done properly. Of course, that’s a really big IF.
Typically, Emergency medicine is done in the Public system: they are geared for acute cases. Once admitted, the patient will either stay in the public system or be transferred to the private system if he wants to. I have never transferred when I’ve been acute: there’s no point. The standard of care is just as good in the public system, and while the public hospitals look more like high schools than hotels, if you’re sick enough to be in hospital you’re not going to notice the plush carpets and tasteful decor.
I usually have surgery in the private system, just because I can. Some of the best vacations I’ve ever had have been in the Southern Cross Hospital, recovering from surgery.