What happens at school side? Does NASA show up with a van to transmit? Are recordings made of the conversations? Is there any way to pickup the terrestrial transmissions?
Contact is either direct by radio i.e. an amateur radio transceiver and antenna set up at the school, or contact via "telebridge", which is a speakerphone-type arrangement set up at the school and connected to an external amateur radio station via the telephone network, and that amateur radio station will establish direct radio contact to the ISS.
Contact is often via telebridge because it is considered more reliable than setting up an amateur radio station at a school, which may be prone to unforseen problems and equipment failure, resulting in the loss of the scheduled contact (and a lot of disappointed people). In some cases the parties involved set up both systems, with radio the primary and telebridge secondary - just in case.
You wouldn't hear the terrestrial - radio - side of the conversation unless you were in VHF range of the amateur radio station (at the school, if a station was set up, or the external amateur radio station if a telebridge is being used).
See section titled "Radio Contacts with the ISS":
About - ARISS