Sounds like the girl has made it up, based on nothing more than perhaps spotting an email with your daughter's name as the title or your name as the sender, or not even that.
I have a huge problem at my school, none of the teachers has an office to work in so we work at our classroom computers. During breaks and lunchtimes the students wander in and out of the room and can easily see what is on our screens, even when they're told they're not supposed to and get sanctioned for being seen trying to look at teachers' computers. In my classroom and most other teachers', the screen of the teacher's computer is visible to the students so they could even quite innocently see things about other students just by walking into the room at the wrong moment without actually trying to look. I hate it. Also we have to be really careful with what we have on our desks, e.g. if it's lunchtime and I'm marking exams., I can't spread out the papers as I'd like to as students could walk in at any moment.
And students like to start using teachers' computers if ever we go out of the room, which, again, is "forbidden", but never really sanctioned properly, and drives me up the wall. I'll be in the middle of lesson planning, pop to the loo, come back and a student has plugged in headphones and is watching YouTube!!!!
Sorry, started ranted off topic. But your daughter's teacher may have tried very hard to be discreet and private with her emails but not actually have a fully secure place to work away from the students.