Who wins? In my opinion...Bradley. Best advice I can give you on this topic...read Natural Childbirth the Bradley Way by Susan McCutcheon.
With my first pregnancy, we attended the "Lamaze Plus" class at the hospital where we would be give birth to our son. This class was their most intensive labor class available. It was 3 hours, once a week, for 6 weeks. Our childbirth educator was a doula and had worked in the field for 20 years. She, herself, had delivered 2 babies naturally in the hospital setting. Sounded good to me. The class cost $100.00 which I think it pretty average for an intensive Lamaze class. Here is the description of the class from the hospital's web page:
Childbirth Preparation – Lamaze Plus
This class is for women who are interested in preparing for a birth experience with less medical intervention. This six-week, 15-hour program includes all information in our childbirth preparation series and allows for additional practice of labor comfort strategies, partner support, and in-depth discussion on working with the natural process of labor. The emphasis is on Lamaze labor and birth practices which support your personal plans for birth. The instructor is a nationally certified Lamaze childbirth educator and doula. A childbirth center tour is included. Complete this course five to six weeks before your due date.
Sounds great, but as we learned, this, and any class offered by the hospital has a definite goal of preparing you for the routine interventions in a hospital birth. Notice the fist line..."for a birth experience with LESS medical intervention." We didn't catch this the fist time around, but having read McCutcheon's book during the second pregnancy we had a new take on our Lamaze experience.
Now, the good side of a hospital taught Lamaze class...you get to see the hospital (but you could do that without the class, just show up, most L&D floor nurses are happy to let you look around), you are exposed to the medical devices used in common hospital interventions like the internal fetal monitor and vacuum and forceps so these devices wont look so scary if you do need them, and it is better than NO childbirth education for sure.
However, the method of Lamaze itself was not very helpful for me during labor and delivery. In the study "Maternal and Fetal Outcome of Lamaze Prepared Patients", it wad found that only 1% (5 women) gave birth without drugs.
Meanwhile, with the Bradley Method, "Of over 200,000 Bradley®-trained couples nationwide, over 86% of them have had spontaneous, unmedicated vaginal births." The goal of this method is to have an intervention-free labor and delivery. The method focuses on relaxation and paying attention to the different signs/stages of labor. Bradley classes are not hospital affiliated, they do not have to purport the agenda of the hospital. The down side of the Bradley method...it costs significantly more to attend a Bradley Childbirth Class series...usually somewhere in the $300-350 range.
However, since my husband and I had already learned to work together during my first labor, I decided to study the method alone, without attending a class. I got the idea from two friends who had done the same thing, one had a successful natural delivery, the other was a planned c-section in the third trimester because she was found to have a septum in her uterus and the baby could not be turned from the transverse position in which she was lying. Secondly, my husband was not "into" the husband coaching thing, so any method I used I was going to have to focus on educating myself mainly. I read both Dr. Bradley's book Husband-Coached Childbirth (pretty antiquated when it comes to his views on women in society) and Susan McCutcheon's book Natural Childbirth the Bradley Way (a much more instructive book on the method itself).
When it came "crunch" time for Audrey's birth, the Bradley method worked for me. I focused myself inward on what I was feeling and relaxing through the contractions. In the slide show of her birth, you'll notice my eyes are closed most of the time during contractions, that was because I was focusing on relaxing, not tensing up when I felt the pulling of the contraction. Todd had read a few chapters of the book, but knows that I'm not one who likes to be told what to do, he was a great partner in the birth, listening to what I asked him to do to help me.
Another good idea when thinking about a doctor is to ask if they are familiar with the Bradley method, since the idea is pretty much that the doctor is only there to catch unless a problem arises. Some physicians are offended by this view and others will say, "oh, right you just want me to catch"...that's the doc you're looking for if you want a natural birth! :) There's probably more I could add on this topic, I'll revisit it again I'm sure.