These are the conversations one can have with allies who are not currently in your party and randomly appear in dungeons that have already been cleared. The allies always appear in groups of two, except for the fox which can appear by itself or with an ally. The contents of the conversations depend on the Social link rank of the person you're talking to, the identity of the other ally in the same room, and occasionally on whether you are in an intimate relationship with one and/or the other person.
There exists a text dump by Diovanti that contains all the dungeon conversations and also some unused lines for reversed S.links that didn't make it into the finished game. Eventually this info should be converted to tabular form here.