The problem you are facing is that the catalog cannot reside on a network drive. The catalog must be on a hard drive that is local to the computer. The images can be on the network drive. Some people have solved this problem by putting the catalog in a Dropbox folder. However, only one user can have access to the catalog at any given time. And you both need to make sure that you allow ample time for Dropbox to update after one of you closes the catalog. The short answer is that Lightroom is not really conducive to multiuser connections. The best (most reliable) way to do what you want to do is have the catalog on a movable drive that can be switched from one user to the other. That one catalog can access the same images on the same network drive.
I'll add a few things to what Jim said. My understanding is you have two computers each with their own separate Lightroom catalogs. All of your edit changes, metadata, and previews are stored in the Database files( .lrcat and previews.lrdata) they are not stored in the images themselves. Therefore, your catalog files are different on the two machines, and this is causing the problems you're having. Syncing the image files themselves won't do any good, because the information you need is stored in the .lrcat file not the images.
There is an option to write Metadata to Files. You can find it in the Catalog Settings preferences. This will allow the images to carry metadata changes, develop settings, keywords and sort flags.
This option however will not include updates to previews, information such as Virtual Copies or other catalog operations. You could replace the LR database files from one machine to another and this would bring the catalogs in the sync, but it has to be done when you shut down Lightroom on the active computer and replace the old catalog files on the other machine with the most recent catalog files from your last session.
This is a tedious and error-prone method. The bottom line as Jim said, Lightroom isn't really designed to be used in a network environment. Jim's final suggestion that you store the catalog files on an external drive that can be swapped between the two computers is the most elegant option for what you're trying to achieve. With this method, you're only using one catalog file and not having to sync two together. The catch is only one of you would time can work in Lightroom. If you decide to go with this option, I highly recommend you use a solid-state drive for storing the catalog files.