Zimbra multi-server is not a master:slave topology, so it's hard to tell what configuration you have truly.
What you are seeing in the Admin Consoles however is consistent with having installed the Logger package on both servers.
If so, this is not correct; there can be only one logger host in a Zimbra multi-server farm.
To fix, I would rerun the installer on one of the servers to remove the logger component, and then update that server's syslog configuration as per the
Multi Server Installation Guide.
This
thread has some dated but detailed information which may be helpful understanding how multi-server syslogging works behind he scenes.
If stuck, you could always uninstall the logger package on both hosts, then reinstall it on just the desired logger host.
Hope that helps,
Mark
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