Basically the owner of the copyright would send a request to the site owner to have it removed. Just like YouTube. However, this rarely becomes an issue with content not related to games. For instance millions of you tube videos use music without permission, but it's not seen as worth the trouble to attempt to get the video removed. Especially when the person is making no financial gain from the use of the media and the actual circulation is limited. In the end there is no real course of action. Generally it's ignored, unless it is a massive distribution that financially damages the copyright owner there is no benefit to pursue legal action.
Yes; there are 3 approaches that i can think of that differ slightly. To keep using YouTube as an example...
1) The owner (or representative) (or someone illegally posing as one of the two) can send an informal request to the host/middleman/service, which may or may not be honored. YouTube has special contracts with a lot of content companies to provide this service, which is why there's no effective appeal process for those. If a company decides it doesn't like your YouTube video, you're screwed.
2) If the middleman has posted a DMCA contact, the owner can file a proper DMCA complaint. The host is then obligated to block access to the content unless and until you file a counternotice, and i'm not sure where it goes from there. There's an appeal process, at least, so you may eventually get it restored. If the host has a DMCA contact, the owner may be
obligated to go through the DMCA notice process instead of suing you directly; i'm not sure.
3) Finally, the owner can subpoena the host for your contact details, and then sue you for copyright infringement by public performance of their property. They can also sue the host if they don't qualify for DMCA protection.
Unfortunately, with game streams and videos becoming more widely known, the common practice of ignoring it may soon come to an end, especially where music rights owned by the music mafia comes into play, such as 80's rock and whatnot. I am fully prepared to rake any game publisher/developer who engages in that over the coals, however. For what it matters. I'm only one consumer who buys a lot of games.