"Would the fax software you use to access the faxes be accessible by all computers on the network?"
This is somewhat of a vague question. By definition, a networked fax server will allow faxing out by any computer on the network. You don't necessarily need fax software on each computer (only the fax driver), although as mentioned before, some of the MFPs and other fax servers load software with the driver to help with the overall process, e.g. bring up the address book (which I know is important to you).
To bring faxes to your PC via a MFP, you need a fax server installed in the MFP. This is generally behind the scenes, and you aren't even aware it is there.
The simple answer is any file that is sent to a server folder can be shared and, therefore, accessed by any computer on the network. This is referred to as a file server, which is the whole idea behind a central server (although you can share in a peer-to-peer as well, or you wouldn't be able to access AC,etc.
On top of this, there is document management softwarer such as Paperport or others mentioned above and SnappyFax that make the files not only accessible but allow the user to work with the faxes in a more robust way.
But, in short, forgetting about any "software" once the document is sent to your server or scanned for that matter, you can decide to share it or not share it with any client on your network.
