I personally use Calibre+Calibre-web.

It’s configured as a proxy for the Kobo store, the default store for my e-reader.

That means that when I click the sync button on my Kobo, it downloads anything hosted on my calibre-web server, while still keeping the ability to browse the Kobo store.

For linux only, lan only shared drive NFS is probably the easiest you’ll get, it’s made for that usecase.

If you want more of a dropbox/onedrive/google drive experience, Syncthing is really cool too, but that’s a whole other architecture qhere you have an actual copy on all machines.