https://syncthing.net/

"Syncthing is a continuous file synchronization program. It synchronizes files between two or more computers in real time, safely protected from prying eyes. Your data is your data alone and you deserve to choose where it is stored, whether it is shared with some third party, and how it’s transmitted over the internet."

No, Syncthing isn't a self hosted cloud but a real p2p transfer tool.
There are discovery servers enabling devices to find each other but files are rarely transferred through them.

Not exactly, nextcloud is more akin to Dropbox where as syncthing just keeps the folders in sync so if you delete or move a file out of the folder the file will be removed on the other machine(s).

But it is peer to peer so you can create a folder, copy your file in and someone else can sync with you using the unique code for your share. The whole thing is encrypted and nobody else can read or intercept the data and it also works great even if you need to interrupt and resume it later . when the transfer has completed if the recipient moves the file it will be deleted from the original folder on your machine. So thats why its best to create a copy to share.

You can do read only shares etc but the official docs go into more detail there