Since you tagged your question "Debian", here's a quick recipe that should solve your problem.
On your lost and finder computers run the following (as root, or use sudo as/if necessary):
# apt-get install avahi-utils
Then, as long as the lost and finder computers are on the same broadcast domain (i.e. the same physical network), you should be able to use avahi-resolve to resolve the IP address (either v4 or v6) of each host from the other.
For example, any user on finder can resolve lost, as follows:
$ avahi-resolve -4 -n lost.local
lost.local 192.168.56.102
$ avahi-resolve -6 -n lost.local
lost.local fe80::a00:27ff:fea7:4900
Similarly, on lost:
$ avahi-resolve -4 -n finder.local
finder.local 192.168.56.101
$ avahi-resolve -6 -n finder.local
finder.local fe80::a00:27ff:fe72:804
Obviously, I have glossed over a lot, and a few little details could go wrong that prevent avahi from working correctly out of the box, but from my own experiments on a few different occasions, there's a good chance that it should all Just Work (tm).