Try restoring the database using the following query:

restore database "dbname"

from disk = 'Servername\sharedlocation\db.bak'

servername - This is not the local server name, it is the servername where the backup files are located.

or else to do it through GUI:

Know the exact shared location, you can type the servername(\\servername) in the "run" and that would autoamtically display all the shared locations on the server to which your account has access. From there navigate to the correct location and use it in the restore path.

The other server need not have SQL installed on it. Its just storing the backup files.