For hospital or doctor/dentist visits, one usually one has to pay out of pocket and then fill out the appropriate forms for reimbursement later. I have never heard of a US insurance company directly paying a European health care provider.
That said, unless your HMO guarantees that they will cover medical expenses incurred abroad (and I would get this in writing with the details of coverage thoroughly outined), you should think about purchasing an insurance policy that will cover you overseas. I spend a great deal of time abroad and ALWAYS do this to make sure that I am truly covered for any possibility. The website I use is insuremytrip.com. It lists many companies that offer trip insurance of all kinds. To get started you fill out basic information about your trip, it gives you companies that are appropriate for your kind of trip, and then you can choose specific options that you want covered. I usually end up going with Travel Guard but there are many other companies listed as well.
Specifically I always make sure that I am covered for trip cancellation, lost luggage, medical or dental expenses including ambulance and emergency room visits, medical evacuation, and (god forbid!) death and repatriation of remains. We had an ambulance ride and ER visit in Italy and the cost was far, far less than here in the states so that time I didn't even bother to get reimbursed because it wasn't bad at all. What really can be expensive is medical evacuation - it can cost as much as a new car or more. And before you say that you'll never need that, I know several people who have almost been bankrupted by this when a loved one had to be evacuated back home via medical jet.
I know that my HMO does not reimburse for medical expenses incurred overseas...really make sure you research this. Maybe yours does but I haven't heard of this before. I could just be in the dark though too.
