Hitbox.tv's terms of use require that streamers own or have permission for content used in their streams, but all responsibility for violation is disclaimed onto the streamer.
Streams are temporarily saved as "Recordings," which you must edit to create "Videos." Videos are permanent, but you can't further edit or take clips from them. Recordings are deleted after 14 days, at which point anything you haven't explicitly saved is lost.
Google's enforcement is even more harsh than Twitch.tv's: streams are not just muted, but automatically terminated if copyrighted content is detected by automated scans.