Sorry about the bugs. I think they're caused by you not having user-level write permission to the folder which Shadows is in. As it is just an Indie game I didn't want to start writing to AppData and making folders on the user's disk, so saves its saved games to where the .exe is. We've now put a check to see if it has permission, and not fail catastrophically if it doesn't. Sadly, it won't be able to save under those conditions.
Regarding the network thing, that's a Unity built-in which we can't disable. It doesn't need any network capabilities and doesn't communicate over the network in any way, so go ahead and lock it down.
New bugfixed version is out, hope it works for you. You may need to disable autosaves in the options (buttom right) or it'll try to autosave once before realising it can't.
Thanks for the feedback