Wasn't it LimeWire/Kazaa? My early P2P history is a little fuzzy.
The big issue is that so many people were still on modems back then that downloading a DC game wasn't easy. Plus public WiFi wasn't really a thing yet. It was the kids on college campuses who were connected to brand new and shiny Ethernet networks that really went nuts.
These applications weren't really where piracy derived, these applications along with a cable modem made it possible for point and click piracy. Similar to Napster. The scene which as others pointed at actually started in newsgroups and shifted to IRC and FTP was where piracy was extremely prevalent.
Especially so, given that the BitTorrent protocol hadn't even been designed when the Dreamcast was officially discontinued.