Just like tv and radio. If they are on screen and not being compensated while the streamer is, it could violate right to publicity. Now that handles and screen names are relevant in society as actual names and identifiers, they will eventually be used in courts as aka.

So would a way to get around that be like censoring basically every username you see? Or messaging the players about the video and asking them if it's okay to include them in the video? Im confused on how streamers an youtubers could get around this

Censoring or blurring would be a start (and will work for post made material on YouTube at least) but since it's live, would be hard to do. Brodcasters have tools and staff in place for their media. It's hard to tell what is needed or what is appropriate. Internet and digital law is still in its infancy. So cases like this, where you don't sue for monitory gain, are important to have a citation to reference when future problems arise. It's how tv and radio evolved and its how Internet tv will as well