monozygote I have only scratched the surface of what is possible with ssh, at the moment I am satisfied with a proper key exchange authentication instead of passwords. Even for that most simple feature, my limited experience is that the configuration would vary between linux distros and os (e.g. PAM enabled or not). PAM would be required for 2FA authentication for example, if that suit your threat model.
I would suggest that you have a look at what ssh can do and whether that suit your need. I don't think it's a definitive source of information, but https://www.ssh.com/academy/ssh/openssh might give you some inspiration, even though the entire website is not 100% dedicated to openSSH. For example you could limit the scope of a user, only allow certain commands, or automatically run a command post-authentication and nothing else. There are ssh server for android but I don't know how robust they are.
I think each user would need to execute its own script to sync its data, which might be an issue if background use is disabled. Rsync or not, there are likely more options that I can imagine, rsync likely being the simpliest of all for automated sync. A basic index files with the sha256 hash of each file to keep up to date (and only copy the new/modified files) would be easy to implement, if that's even a requirement.
At that point a properly configured NFS or equivalent with strong authentication would probably make more sense. Ultimately your data will have to live somewhere to be synced. I am not there yet but it doesn't have to be a centralized server with 24/7 availability, it could be a vm/container that only run when needed to distribute its data to the various endpoint. Could be automated, manual, or any combination. Or even something like Nextcloud, which seems to be quite popular. Something on my to-do list to try. Personally, if I had a need to sync like that on regular intervals I would likely add a VPN connection and still rely on ssh or tls, to only access the data on a specific VLAN, or something of that order.
A non-centralized approach is certainly possible but I would recommend that you put it on paper and determine how it will be managed. "Knowing" which file is the most up-to-date is certainly an option but it might stretch the definition of syncing a little.