Mute notifications whenever a screenshare is detected (assumes you use dunstctl / org.freedesktop.impl.portal.ScreenCast), but can be adapted.

mute_notifs.py

#!/usr/bin/env python3
import dbus
from gi.repository import GLib
from dbus.mainloop.glib import DBusGMainLoop
import threading
import os

def main():
    DBusGMainLoop(set_as_default=True)
    bus = dbus.SessionBus()
    obj_dbus = bus.get_object('org.freedesktop.DBus',
                              '/org/freedesktop/DBus')
    obj_dbus.BecomeMonitor(["interface='org.freedesktop.impl.portal.Session'", "interface='org.freedesktop.impl.portal.ScreenCast'"],
                           dbus.UInt32(0))

    bus.add_message_filter(msg_cb)

    print("Started up, listening on dbus for sessions")
    mainloop = GLib.MainLoop()
    mainloop.run()

def msg_cb(bus, msg):
    member = msg.get_member()

    write_opt = None
    try:
        if member == 'Start':
            print("Detected started screenshare")
            write_opt = True
        elif member == "Close":
            print("Detected end of screensharing")
            write_opt = False

        if write_opt is not None:
            # Don't judge me, you try doing a dbus call in another thread without
            # python complaining.
            os.system(f"dunstctl set-paused {str(write_opt).lower()}")
    except Exception as ex:
        print("Encountered an error while trying to set the paused flag", ex)

if __name__ == '__main__':
    main()