Does My Photo Stream work without an Internet connection? (local wifi only)
To the best of my knowledge, Photo Stream will work whenever it is connected to iCloud.

It doesn't matter if the connection is via your carrier data plan or via wifi to the internet. But you must in either case be able to connect to the Internet. 

Of course, you must have cellular data turned ON on your device in order to connect to iCloud via cellular.

Without the Internet connection to iCloud, you will not be able to use Photo Stream simply between your devices, even if they are all on the same wifi network.