When I've installed motion in the past it's been done so that it's running as its own designated user, typically motion. I'd suggest doing the same thing here for your installation as well.
EDIT #1
The OP asked how this was done. I explained that if you install the motion package via either Debian/Ubuntu or Fedodra repos the installation would be done so that everything you needed to run motion as another user, motion in my cases, was done out of the box by default.
If you look at the files that would typically get installed with motion, a SYSV init script is often provided, /etc/init.d/motion. Within this script, on Ubuntu, is a section like this:
case "$1" in
start)
if check_daemon_enabled ; then
if ! [ -d /var/run/motion ]; then
mkdir /var/run/motion
fi
chown motion:motion /var/run/motion
log_daemon_msg "Starting $DESC" "$NAME"
if start-stop-daemon --start --oknodo --exec $DAEMON -b --chuid motion ; then
log_end_msg 0
else
log_end_msg 1
RET=1
fi
fi
;;
If you look at the start-stop-daemon line you'll notice that when motion is started ($DAEMON) the switch --chuid motion is passed, which will run the motion daemon process as user motion.
Something similar is done on my Fedora & CentOS systems as well in their corresponding /etc/init.d scripts for motion.