To backup all data on a local computer you need to at least be a member of the Backup Operators Group on the local computer. To restore data on a local computer you need to be a member of the Local Administrators group.
Users don't require any special permissions to backup files they have read, modify, or full access to. If a user only wants to perform a backup of their own files, they don't need any additional permissions.
To maintain security, only Administrators and specified backup operators should have access to backup all files on computers that require backups