Hi JellyBean. I planned to have my twins naturally, and did a lot of research on the subject. Unfortunately, I was faced with an emergency c-section due to fetal distress. One of the boys wasn't breathing at delivery, so it was good they intervened when they did. But overall, most of my research said it was reasonably safe. My boys were monochorionic, so there was some risk of twin-to-twin transfusion syndrome popping up. Besides that, in general, the risk with delivering twins is more with the second baby, not the first. You need to be mentally prepared that you could vaginally deliver the first, and then need an emergency c-section for the second. It isn't common, but it isn't extremely rare either. And if something goes wrong, it will happen very quickly. They only have a few minutes to take the baby once he starts showing distress.
The most risky presentation is if the first baby is breech and the second baby is vertex. In this presentation, the babies necks could get stuck on each other and it is very dangerous. This is the only presentation that my doctor would not deliver vaginally.
One of the big decisions you will make is how much monitoring you will be comfortable with, especially for the second baby. Because the delivery of the first can cause distress on the second, many practitioners want to monitor the 2nd baby by ultrasound during delivery of the 1st, which limits the birthing positions. But even for the 1st baby, because of the slightly increased risks, some practitioners want to do internal monitoring, which can restrict your ability to move around.
My best advice is to find a doula who has been present at other natural multiple births.