Can you save to desktop first and then move that saved file over to your network server?
Yes, that's not a problem, but it is inconvenient.
I can save ID, Acrobat, AI files all to the server just fine, export images. It's just saving out of photoshop as anything except a PSD file that won't save to the server.
This is the boilerplate text often used in connection to saving to a network (please NOTE the part where it explains that normally, it does work, but that it is impossible to troubleshoot someone else's network remotely, and that's why it's not supported by Adobe):
If you are opening files over a network or saving them to a network server, please cease and desist immediately in the event you are currently experiencing problems with one or more files. Working across a network is not supported.
Copy the CLOSED file from your server to your local hard disk, work on it, save it again to your local hard disk, close it, and copy the closed file back to the server.
Of course, the fact that Adobe does not support working across a network does not necessarily mean it won't work. It should.
Adobe's position is that there are too many variables in a network environment for them to guarantee that everything will work correctly in every network, especially given the fact that if something does not work properly, it's probably the network's fault, and Adobe has no way of troubleshooting your network.
If you can't work locally, you are on your own, and if something happens, you're on your own. If you must work from a server, make sure your network administrator is a competent professional.
When problems arise, a lot of valuable work can be lost.