I have found btsync (now Resilio Sync) the best available p2p solution, albeit requiring sync app installation, still you need some app anyway, being it the browser, file manager etc..
It uses local network when available and of course internet when not. It also does iOS.. if only more apps were supporting FileProvider functionalities, so you don't have to import-duplicate all the synced data first.
It requires installing proprietary software on all devices though, while your browser already brings along an open source and mostly standardized P2P stack.
There are open source alternatives like syncthing, which is still not as user friendly.
Also single file sharing seems interesting, but this is not what you usually do on daily basis... You just want to drop some file to shared folder most of the time, or to have a ton of different files synced automatically.
I've tried syncthing as well as btsync (now called resilio); it's not just less user-friendly. It tends to be _much_ slower and eat tons of CPU due to a different algorithm. Resilio has a free tier that works fine for me, and I don't have an inherent issue with a proprietary program that works well. It's a very convenient solution, I haven't had issues, and highly recommend it.
It uses local network when available and of course internet when not. It also does iOS.. if only more apps were supporting FileProvider functionalities, so you don't have to import-duplicate all the synced data first.