I happen to run a Free Clinic in NC (I'm an NP), but also have done the mission stuff in Peru too. Depending on where you live, some Free Clinics have gotten coverage through the Federal Tort Claim Act so that if you were sued, they really have to sue the Federal Government (kind of how they do at Military facilities etc.) We haven't gone through all the paperwork for that yet but we haven't had any problems yet either. Our clinic uses 150% of the Federal Poverty level as our guideline and we make the patients "prove" their income as best we can. We also get their meds through the Patient Assistance Programs that the pharmaceutical companies operate.
I have found that my patients run the gamut...some really are very poor while others just haven't quite gotten their act together to get a job etc. Some are very thankful and others feel like they are "entitled" to our services even though we don't take a dime of state or federal money....everything is funded through donations, grants and volunteers.
One interesting note about foreign mission work...I have gone on a specific trip to the same place in Peru twice. Once in 2005 and once this past year. The first year I went most of the moms and kids were fine---mom just wanted you to "lay hands on them" because they lacked medical care and wanted to make sure they were ok. This past time, however, they have gotten a bit "spoiled" with our coming every year. The moms would complain that their kids didn't eat, or were sick....even though you could tell that they were most (not all) quite healthy--- just because they were so accustomed to us giving out children's chewable vitamins, toothbrushes etc. etc. and maybe they were afraid we wouldn't give them anything if we told them there was nothing wrong. It would make me laugh because they would say "oh, my son is so short" when mom was 4-10 and dad 5-0 tall.