Turns out: NO ONE KNOWS.
It's pretty much the consensus that if you're flushing into a septic tank you're going to jack it up with your lady business, but the jury is out on other types of plumbing situations. And to be fair, it does say on the tampon directions that you should dispose of tampons through flushing. There's a lot of info out there, but the only concrete conclusion I can fully commit to is that there are some people out there who are passionate about the anti-tampon-flush crusade, and there are many who, like me, had no idea there was an issue with flushing tampons. I'd recommend doing some reading here (the most interesting/entertaining commentary I found). That is, if you're as interested in period politics as I am.