As far as I understand you (it is not that clear), the Trisquel system has
two network interfaces. An Ethernet interface and a Wifi interface I guess
(all laptops and some desktops have that). You can connect the Trisquel
system with the Ethernet cable and then the other system (have you tried
Parabola GNU/Linux, which is based on arch and 100% free?) connects through
Wifi. To do that, the Trisquel system must act as an access point. There is a
button for that at the bottom of the "Network" window (launched from the
"System settings"). I used to do that some versions ago and the package
"dnsmasq" had first to be installed (from the "Synaptic package manager" for
instance).