The "USER" and "DISPLAY" variables will be part of her environment.
You can grep out the PID of a process she will be using, and then
tr '\0' '\n' /proc/$pid/environ | grep DISPLAY=
I had just posted this, and other things, in your other semi-related post on cron, before reading this thread.
Look at the Xstartup script, if you have it. It does things like parse the DISPLAY variable to tell if it is a local or remote connection.
Code:
# Find out if this is a local or remote connection
#
LOCATION=${DISPLAY%:*}
LINE=:${DISPLAY#*:}
if test -z "$LOCATION" ; then
# local connection
case "$LINE" in
:0|:0.0) LOCATION=console ;;
*) LOCATION=localhost ;;
esac
else
# TCP/IP connection (remote or local)
NAME=${LOCATION%%.*}
if test -z "${NAME%%*[0-9]}" ; then
LINE=${LINE}/${LOCATION}
else
LINE=${LINE}/${NAME}
fi
fi