Is it cool for River to read your character's mind? Yes; however, he's also carrying twenty or so demons around with him in confinement tubes, with one or two of them free-roaming, so she may find more than she bargained for.
Is it cool for her to talk about it? Yes.
What might she find there? For Raidou: lots of worrying about various people/the Capital/his duty/his demons/camp. Some ruthlessly cut off thoughts about Narumi (lol unrequited crushes). Currently traumatized from the card event.

The demons, if she can read their minds, will vary widely, although Gdon will be very focused on Raidou's safety and well-being. Special mention should probably go to Oboroguruma for being an undead sedan with seriously random thought processes, a tendency to spout absolute gibberish, and a fondness for Russian literature.

Would your character notice? Raidou might, since the second game implies that devil summoners are trained to deal with psychic intrusions. The demons undoubtedly will, and several of them (Inugami, Lilith, Beelzebub, Black Frost, Mara) are mind-readers themselves, so she may get responses of varying politeness, and they will want to ascertain whether she is a threat to Raidou as well.