Two new great tweaks are supposed to start rolling out in February.
This is actually two settings described in one roadmap item. Some time ago, Teams introduced chat bubbles for users who closed their chat panel. Now, users get the options to mute these bubbles through a menu in the meeting. To enable this, just check the "Don't show chat bubbles" item. This is a per meeting setting.
The other option is to globally (per user) mute all notifications whenever you are in a meeting. This will also mute the chat bubbles. It will also mute all chat and other Teams related notifications from outside the meeting.
Another nice tweak coming soon, is the capability to hide your own video feed in meetings. This doesn't stop your video, but simply removes yourself from what you see on the screen.
None of these features will need any admin actions, but user manuals and training should be updated to make users aware of the new features.