Most advanced countries have surveillance in place for all medical interventions. In the UK we have the "Yellow card scheme". Most submissions are by medics but patients can write up their own reports too. Lying doesn't really matter because they're looking for patterns, so a few people writing nonsense doesn't trip the process up (but of course, please don't lie).
However, we don't have good baseline data. One thing that surprised me was, for the HPV vaccine which was initially being given specifically to teenage girls, they didn't have baseline menstruation symptom data. So, does having the HPV jab tend to make your period late? early? heavier? How often? We have no idea because we had no baseline data. Maybe the reports you get are actually normal - prompted only by the recent vaccination. Or maybe they aren't. If you were going to pick a group of patients likely to freak out because of minor and inconsequential menstrual changes you would pick teenage girls, but nobody had this data. Not just in the US, everywhere doesn't have this data. For easy stuff like "Do people drop dead when they take this vaccine?" we know long before authorisation if there's a big problem. But for subtle effects it's invisible until surveillance and without a baseline we have to guess what's important unless the numbers are overwhelming.