Different sites use different means. You can create a throwaway email address, but it's just one more data point.
> Your example is a very public lawsuit by the MPAA against a public tracker?
If the site operator agrees to keep the site up while data is collected there is no court order. Every site operator who gets raided ends up facing some legal consequences but the details of the deals they work out with the industry are not made available for the public
> Give an example?
https://www.cashmanlawfirm.com/list-of-strike-3-holdings-cas...
Outside the US lawsuits continue as well (examples): https://www.mosshacheylaw.com/blog/movie-producers-suing-can...
https://www.canadianlawyermag.com/practice-areas/intellectua...
https://torrentfreak.com/torrent-site-user-who-transferred-1...
> They won’t demand money from you unless you’re making a profit.
Not even remotely true. Please don't assume this. I'm aware that they have made exceptions in cases where people were literally impoverished but fines are typical in settlement offers.