Are you running the first PC as a server and the second as a client using the MC Media Network?
That is the way you should be running it, so that there is one library on the server, which is then sync'd to the library on the client. If each PC is operating independently and connecting to the media on your network location, then each PC would have its own library, which you would need to maintain separately. That should work for tag updates, as long as the tags are stored in the media files (if they support this) or sidecar files and not in the library. Even with sidecar files, I'm not sure MC would detect changes and do the updates in the second PC library. So make sure you use the first PC as a server and the second PC as a client.
Do you have "auto sync with server" set on the client PC under Options -> Media Network -> Client Options (when connected to a Library Server)?
With this setting active, changes in the server library will be synchronised to the client library, so you will see tag updates on the client. The sync can sometimes take a little while. If you wish to check tags on the client immediately, and it was open when you made the changes on the server, on the client you can right click on the library you are using under "Playing Now", "Playing from" and select "Sync Changes with Library Server" to force a sync. That option is also under File -> Library on the banner menu as well. Closing the client instance of MC, and reopening it will also force a sync, or disconnecting from the server and re-connecting.
I'm not sure what you mean by "assign saved DSP values", but if you mean DSP settings under Options -> Audio, then I don't think they are synchronised, as they are dependent on the PC that MC is running, either the Server or Client. They need to be set and maintained separately, as there are separate output devices on each PC, which require different settings.
Any tag changes, and the results of audio analysis, will be sync'd to the client though. Of course you need to have "Write files tags when analysing audio, getting cover art, and applying folder-based tags" set on the server under Options -> Library & Folders -> Auto Import -> Configure auto-import.