It's a bit tricky, because the grammar isn't quite "strict" :-)
hearsay is a rumour, something that is repeated without an identifiable source ("they say", where "they" is unknown, something you just heard around).
So, a "hearsay she say twist", can be interpreted as a twist on that rumour, when it goes from "hearsay" (no identifiable source) to "she say(s)" (she is now the origin of the rumour) - that's the twist.
"Gotta put a mouth to it" repeats the same meaning: must put a mouth to the rumour, now "she says it", instead of being hearsay.
"just a little hearsay" - a rumour
"they say shit" - they speak nonsense
"I'm down with it" - I'm fine with it.
Remember that the whole gist of the song is about a relationship where the trust has gone, and one person has lied to the other. So, in this context a rumour can be destroying what's left. At the beginning of the song she says that she'd like to go back to when it was just for fun. And I imagine she's done something and her partner has heard a rumour and doesn't trust her anymore?
Sadly, like most pop songs these days, the lyrics are catchy, but the meaning is very unclear, and words are just chosen because of they way they sound, rather than because they make sense.