What is the minimum rights required to access the Users directory on another computer via an admin share?

I have a batch file that writes some information to a few other computers using a path of \\%COMPUTERNAME%\c$\Users\%USERNAME%\AppData\Roaming. The batch files run under an unprivileged user (part of Domain Users only). How do I set appropriate rights so that service account can access the AppData\Roaming folder for every user on another computer?

I'd like to give rights lower than Local Admin, which I know will work.

Things I've attempted:

1. As Domain Admin, attempted to give Modify rights to the C:\Users\ directory on the local computer. Error: Access Denied.
2. Set the service account as Local Admin on the other computer. This works, but is against IT policy where I work. I'd like to accomplish this with rights lower than Local Admin.

Any suggestions?