The pictures can be on the file server. But the catalog must be installed locally on each computer. Make sure that the network drive has the same name on each computer. This also means that any adjustments made on one computer will not be seen on another. The only way to do that is to use a single catalog that is on a EHD that you can move from one computer to the other. Lightroom is NOT designed as a network solution.
The above statement is True about the catalog has to be on each local computer. It can't be on the server.
But the part about adjustments not showing on other computers is false. That is IF you have the option checked to Auto write changes to XMP then you will see the adjustments on every computer. By default LR store all edits in the catalog file, Dumb IMHO. Checking that option write the edits to a XMP sidecar file which can be read by other installs of LR.
Just one point to clarify. Writing changes to XMP will not cause the view to be updated on other machines. What you'll get is an icon on the thumbnail, with an exclamation point. Hovering over the icon will tell you that metadata has been changed by an external program. Clicking the icon will give you a dialog box with the option to either import the metadata from disk and update your catalog, or to overwrite the settings in disk.
Alternately, you can highlight one or more photos, and choose Metadata > Read Metadata from File...
A clunky workflow at best, but further illustration that LR is not designed to be a multiuser networked program.
This is correct. Lightroom is specifically designed as a single-computer, single-user system.
Most likely the easiest (not prettiest) way to share the catalog between computers is to put the catalog on an external drive and move that drive from computer to computer as needed, as suggested in Jim's response #1.