There have been several court cases so far so this subject is important for both private server owners as well as players. There are rules you have to follow when creating your private server if you don't want to end up in a court. As a player you also need to be careful because Offical game servers might be interested in your activity as well.
This is not something to take lightly. There have been several cases where official game servers such as WoW have demanded millions of dollars from various popular private server owners by court. Official game servers see your private server's profit as a loss, simply because the player's money used in private servers could have been used in official servers instead. There are of course multiple factors you have to take into account that might vary depending on the official game itself.
Make sure you follow these guidelines:
Regarding playing in such servers, the official game servers can't do much about stopping you to play on private servers except shut down the most popular servers. They can however directly influence you if you streamer for example. If you stream your private server gameplay, they can shut down your stream. This means that you're a very popular streamer and have been influencing other players to play on private servers instead. They might not even let you directly now that they are going to shut down your stream. All they have to do it simply explain the situation to Twitch - who will ban your streaming account. Twitch usually does this without a second thought, because they don't want to get caught in a court case.
What if you have been made an administrator or moderator in a private server? In this case the official servers have no beef with you, unless the server is co-hosted with your help. Let's assume your partner creates the server and you create the website for the server then you are still considered a participant in hosting a private server. So those who help in hosting the server are the ones that have to be careful in being compliant with the offical game. This means that merely as a player you shouldn't worry about getting in trouble.
Before involving yourself with a private server, whether you are a player or a creator of one - be 100% sure that your involvement is legal by copyright, EULA and other legal agreements.