I found a post talking about KaZaA and Napster, and that's right: I forgot but Napster had a central server to provide files, that's why it was so easy to shut down...
https://computer.howstuffworks.com/kazaa.htm
KaZaA didn't store the files itself so it was thought they wouldn't be possible to shutdown. From the site above:
"While Kazaa claims to be "completely legal," there are those who disagree: The free-to-download blue files are controlled by Kazaa users and include copyrighted content."
"Later that year, Kazaa was sued again, this time in the United States by the Recording Industry Association of America (RIAA) and the Motion Picture Association (MPAA). As of February 2005, the decision in that suit is still pending."
I remember they started suing individual users at that time... I found an article explaining that:
https://www.videoproc.com/resource/what-happened-to-kazaa.ht...
"In September 2003, the RIAA filed lawsuits against over 250 individuals, accusing them of illegally distributing about 1,000 copyright music files each, using P2P networks. RIAA sought an average compensation of $3,000 per case."
The result of the first case:
"In July 2006, the MGM Studios, Inc. v. Grokster, Ltd. caused Sharman to settle for $100 million, the amount to compensate the loss of four major music labels – EMI, Sony BMG, Universal Music, and Warner Music. The company also agreed to pay an undisclosed amount to the studios in the industry."
It's unclear exactly how Kazaa got down, the article concludes with "In August 2012, the Kazaa website was no longer active."... "the rise of legal streaming services such as iTunes, Spotify, and Netflix further compounded Kazaa's demise.".
Looks like the music industry managed to scare people away from pirating instead of actually succeeding in bringing them down directly, which is more or less what I remember.