I had to go to the hospital in Germany, and they did send me a bill.
I was living in Hungary at the time and had just recently quit my newspaper job there, so was no longer under the Hungarian health insurance. I did not have any coverage from America, either. Long story short, I was on assignment at the scene on an autobahn accident when I tripped and fell on my camera lens, which completely gouged into my eyebrow. I was taken to a hospital in an ambulnace, given a tetanus shot, and got sewn up with three stitches. While I was there, they took my passport and I gave them my address and other personal information.
A couple months later, I got two bills in German in the mail to my residence in Hungary: one for the ambulance, and one for the hospital visit. The total was about $1000. From what little German I could read, the method of payment was supposed to be some sort of European bank draft, but I had no European bank account, and, besides, I didn’t have enough at the time to cover it. So I made a good faith effort to pay them, sent them a letter stating the above, give them my credit information, and said, I give them authorization to take the money from my credit card. If they couldn’t do that, please send me a way to pay by credit card, because I don’t have the resources to pay via a bank draft.
They sent me another letter of dunning a few months later, with no reference to my correspondence. This time, they threatened something with the embassy or another (I have no idea, I’m not very good in German.) Once again, I sent them a letter saying that I want to pay the bill, but I could only do so with a credit card or some sort of payment plan, and once again gave them my financial details.
Whatever happened with that, I don’t know. They never charged my card, and they never contacted me again.
ETA: Oh, and I had gotten sick and visited the doctor or hospital in Hungary several times. The thing is, I could afford it there without insurance.