EDIT: I know you said without going through the internet. Syncthing can be configured to only transfer over specific networks (e.g. home LAN/WI-FI)
I keep my phone's picture in sync with my personal computers ("send only" so I can remove old photos when my SD card is getting full).
I sync a "media" folder where I dump all the music and video I download with the youtube-dl CLI (a "yt" alias makes sure the files are stored in the right directory with some custom parameters).
I sync my KeepassXC databases (work and personal), between my personal Linux laptop, my Android phone and my work MacbookPro. Databases can be merged in a single click if there's any conflict (happens very rarely). I love the Android secure autofill service and fingerprint quick unlock
I use a "temp" folder to drag'n drop stuff between computers so I can file it properly on the right device. On Android, I prefer to use the Syncthing "sharing intent" to make any file/media available on my other devices in just a tap.
I also have installed Syncthing on my Android TV, and occasionally drop HD movies that I download on my phone over P2P (I have a pretty fast connection, so it's easier for me to choose a move from my phone, while in commute, have it download in a few seconds, and either stream it to my TV via Chromecast or open it from the synced folder through Kodi)
This really is a dream setup.
Here is the related issue: https://github.com/syncthing/syncthing-android/issues/29