My question is, is it truly safe to leave children with someone who isn't considered by society at large to be mature enough for many decisions?
The answer is: it depends (I know, I hate that answer!).
There are a lot of factors to consider: How old is your child? How prepared and educated is your babysitter? Are there any medical concerns or other specific needs for your child? How long will you be gone? Is there anyone you can have stop in to check up on how things are going? Can you call periodically for status updates, or have the babysitter text you? How quickly can you get back in case something goes wrong?
All of these can influence just how safe it is to leave a child with a babysitter, regardless of the age of the sitter.
I would be hesitant, at best, to leave an infant, or even a toddler, with a younger sitter, but if my child was old enough to be in school, I might be more comfortable with a teenage sitter.
The closer I could monitor the situation, and the faster I could intervene if necessary, the more comfortable I would be with a younger sitter.
The American Red Cross provides a babysitter certification course that seems at least somewhat reputable (for example, University of Michigan's Health System website recommends it). This course is aimed for students 11 and older (note that even U of M's website recommends against having children under the age of 12 babysit), and they even offer supplemental courses (and certificates) in pediatric first aid and CPR.
However, the study referenced in Joe's answer concludes that while middle school-aged babysitters will likely encounter common household emergencies and therefore benefit from first aid training, very little difference in safety knowledge was found between trained and untrained babysitters. This puts into question the effectiveness of the training.
Are there some sort of statistics/studies on the relative safety of using a teenage schoolkid babysitter vs. a professional adult one?
I was unfortunately unable to find any. That doesn't mean such a study doesn't exist, but I imagine it would be very difficult to find relevant data from which to derive such a study. In order to do such a study, there would need to be a way to correlate safety incidents with training. The study mentioned above does this by issuing surveys, which the participants self-administered. I have less faith in survey-based studies for safety issues like this than ones that rely on medical facility reporting requirements, and I would be surprised if age of babysitter would be included in many standard reports.