First the bad news: a few freestanding birth centers may give pain medication but most don't because of the potential adverse effects on mom and baby. The good news is that it is rare for a woman to transfer to the hospital in labor solely because she wants pain meds--although that will certainly be an option should that be the case for you. Why? Not because women who choose birth centers are somehow different from the common run but because: (1) routine and frequently used hospital procedures (labor induction, oxytocin to strengthen contractions, IV drips) and restrictions (confinement to bed, forbidding food and drink) increase pain, and (2) birth centers offer comfort measures in labor that women themselves rate as highly effective that hospitals rarely do (deep tub immersion, showers, birth balls, rocking chairs) and, as you yourself said, a relaxed, homey environment where staff are supportive and encouraging. Anxiety and stress also increase pain. I also recommend that whether you can get childbirth classes or not, hire a doula. A doula is a woman with training and experience in doing labor support. Several organizations certifiy doulas. Here is an article of mine that will answer some questions you might have and another that will help you choose one.
-- Henci