Is the switch managed?
Is the switch a managed one - one with some sort of administration interface - where you could mirror all the traffic to a given port (also called span port or the like)?
If so you can setup one of the PC (even remotely) to capture traffic during a given period (that should representative of the traffic), and everything will happen at the remote site.
If PC are Windows one, you can use WinPCap + Wireshark (command line tools available), Microsoft Netowrk Monitor or any other tool to capture and then analyze the traffic. Wireshark works on Linux too, but there are other tools as well.
You could also set filters on both capture and analysis phase to see only the outgoing traffic ignoring the LAN one, and then maybe have scripts uploading you files when there is little use of the network, i.e. at night.
If the switch isn't managed, it's much more difficult - you'd need to install additional hardware to intercept the traffic you need to analyze. An hub between the router and the switch (if neither can deliver traffic to another device) can be used to capture the traffic between the devices, but a PC needs to be connected to hub to receive traffic. Inspecting traffic among PCs may be harder, though.