I have a Toshiba Satellite with Windows 10 that connects with a 'privacy' network while the Lenovo Laptop has Windows 7 and connects with either a homegroup or workgroup only. How do I connect them together? They don't seem to work with the privacy network.

You can connect both laptop from cross over cable it will be work on windows 7 laptop. As you told that both are different windows and want to connect from different network

http://windows.microsoft.com/en-us/windows/connect-two-computers-crossover-cable#1TC=windows-7