You could ssh on each machine to detect the firefox process running on your name.
For example check on :
ssh myname@1.2.3.4 ps -ef|grep firefox |grep myname
If you have a list of machines, you can do it in a loop, for example:
for box in 1.2.3.4 1.2.3.5 1.2.3.6; do ssh myname@$box ps -ef|grep firefox |grep myname done
You may use ping to list the machines in your lan (assuming /24 subnet mask), example:
ping -b 1.2.3.255