The DMCA is a US law, yes, but Europe has its own equivalents. Every commercial grid I know of, including those based in Europe, have DMCA-style take down policies in place, or state outright that they comply with the DMCA.
Even China has a copyright infringement law in place -- and US media companies have successfully gone after Chinese movie piracy sites.
The laws might different in specific details -- for example, the US law requires a registered agent with the US copyright office, while most other laws don't have an equivalent requirement. But overall the takedown process is pretty much the same everywhere. The copyright holder or their representative sends a note explaining where they found the infringing content, and affirming that it is, in fact, infringing on their copyright. The content gets taken down. The person who put up the infringing content can appeal. If they do, the content gets put back up. At that point, the original infringer can go to court to fight further. Most infringers, when they get caught, just slink away because in order to appeal the take-down request they have to say who they are -- and give the other guys an opportunity to sue them. Most pirates aren't going to do that! If they give a fake name to avoid being sued then their appeal is invalidated and the infringing content is taken down after all.