We should log locally as done before (or as if syslog-ng isn't installed).
Its badly practice to not also log locally, because what if the centralised logger goes down? (The concern is mostly about having it just inside the centralised logger increases the complexity of resolving issues as we may need to look at the physical log).
Sometimes its easier to just search through the log using tools.
It should keep logging to the usual files like syslog and auth.log and so on, we shouldn't just log to syslog (the file).