To toggle on or off Streamer mode, press the keybind Ctrl + Shift + S, or Command + Shift + S on Macs. You can be in any game screen when doing this. Your in-game name and Battletag will now be censored in matches and at the top of your friends list, alongside hiding all opponent names in battlefield screens.
Your name on the battlefield screen will be replaced by "You", and your opponent's name will be replaced by "Your Opponent".
Activating Streamer mode while in the middle of a game will hide your name and Battletag in your friends list, but it won't hide your name and the opponent's name on the battlefield screen until you leave the current game and queue into another game.
Your preference for whether you have Streamer mode on or off is saved the next time you open the Hearthstone client again.
- Blizzard Entertainment (November 5, 2019). Hearthstone Update – November 5 – Preparing for Descent of Dragons!. Retrieved on 2019-11-27.