software-based dual-monitor solution via LAN

Assuming you're running Windows, MaxiVista can do this:

MaxiVista turns any spare Desktop, Laptop or Netbook PC into a dual monitor for your primary computer. No extra multi monitor hardware is required. Simply extend program windows across multiple screens as if it were one big monitor. Increase your productivity by using multiple monitors.

MaxiVista is not free though.

There is also ZoneScreen but I've never used it and don't know how well it works.

Edit: Actually this question is a duplicate of many questions, as you can see by checking all questions whose answers mention MaxiVista :-) .