I doubt, if handled properly, that you would have problems in most countries. But best to get all the ducks in a row prior to coming. Local contacts are not essential for that, but sure would be convenient. If Hungary is the location, i could point you in the right direction.
My guess is you may need to purchase your meds locally as a nurse may have issues with injecting something not obtained by a local doctors order. That of course means visiting a local doctor with all of your US paperwork.
I am an extended tourist here (well over 90 days) and it's exactly what I had to do. Everything worked out easy. Great doctors, great care, cheap meds. Fortunately I got good at slamming the needle into my thigh all by myself.