I think you're right to research and question, and no vaccine is 100% safe (regardless of what you might be told). However, it's also important to remember that the latest statistics have 25% of teenaged girls in the US infected with an STD--yes, that's 1 in 4, and that's ENORMOUS, particularly when you take into account that some girls don't have sex at all while teenagers, and many others don't have sex until their late teens. So of those who are sexually active, many, many end up carriers of something.
It's also great that you have open lines of communication with your DD about this--I think that's VERY VERY important, that she can come to you even if she's making decisions that she knows you might not agree with (since, after all, they are her decisions, even if she is young...) One other thing to keep in mind, which is not often discussed because it's not nice to think about, although it is true, is that too many young girls don't get a choice in when they become sexually active. This happened to my mother, and as a result she was very proactive about my being on birth control, going to the gyn, etc, when I was in high school. She obviously hoped that such a thing would never happen to me (and, thankfully, it didn't), but she also felt strongly that we need to be prepared for those things in life we can't control. I'm thankful that she taught me this.
I know it's a hard decision, and commend you for thinking and researching and asking advice. Good luck finding the decision that's right for your daughter!