In most cases they are authorized (and paid by) the legal copyright holders to find violators. At least in US most cases are brought by industry organizations.

I heard (and believe it's true) that in Germany, lawyers are given the right to bring cases against law breakers (maybe only civil) with no client, without support or even knowledge of the "victim". Paid on a bounty system. Idea being all these lawyers are incentivized to go out and deter crime.