How to remove the notifications in Pokemon Sword

Unfortunately, there is no way to completely turn off this functionality. The best you can do is make sure you are not connected to online communications, so you will only get your own notifications. This, however, makes it kind of hard to trade or battle online, if you are doing that.

To turn off notifications from others:

• Open the Y-Comm (press Y)
• At the bottom of the screen it will either say "Connect to the internet" or something like "Disconnect from the internet".
• Press the + button to switch between these states. Disconnecting should only show your own notifications.

Other than that, completely disconnecting from the internet (Flight Mode on your Switch, or turn off WiFi) should also disable the notifications for as long as you are disconnected.

None of these are ideal solutions, but until they patch some kind of setting into the game, it is not possible to completely hide the Y-Comm.