Yeah, I use Syncthing for this.
You can add basically any folder, although unfortunately Android seems to have some stupid restriction on using the whole Download folder.
There is a share-menu entry for saving to Syncthing, although to be honest, I've never tried it.
This is entirely optional, but I have it set up, so that my individual devices have a DNS name in my router and then I've told Syncthing to just connect directly to that DNS name without doing discovery or whatever. If you do just want it to sync in your local network anyways, this makes it a bit more reliable / quicker to find all participants.
Syncthing does sometimes stop running in the background, so I have to open the app to make it sync again, but as far as I can tell, this is yet another stupid Android restriction, not something another app can improve upon.