Run a command in another session's X-Windows environment, even under gdm3 where the .Xauthority file is hard to find.

gistfile1.sh

# This can be copied+pasted directly into a terminal window
#  and the display-authority function will become available.
# usage example:
#  display-authority :1 'notify-send "hello, stranger!"'
# Note: this uses sudo access to find the appropriate XAUTHORITY file,
#  so you'll need to be a sudoer to use this.

function display-authority() {
 export DISPLAY=:${1#:}
 PROC=$(sudo fuser /tmp/.X11-unix/X${1#:} 2>/dev/null | sed -re 's/^\s*([0-9]+)\s*$/\1/')
 echo Process $PROC owns DISPLAY $DISPLAY
 export XAUTHORITY=$(sudo cat /proc/$PROC/cmdline | sed -re 's/^.*-auth\x00([^\x00]+).*$/\1/')
 echo XAUTHORITY is $XAUTHORITY for DISPLAY $DISPLAY
 shift 1
 echo Executing command: $*
 sudo sh -c "$*"
}