Bit vague requirement, processes must be running on servers
Using, for instance,
local5.info facility and
logger command inside a shell script.
You configure that servers syslog to send logs to your central syslog server for that facility on IP using UDP or TCP.
Central syslog server will receive / parse etc. the logs messages.
Great choice for server side is any OS that can run rsyslog, which is good software, highly customizable.
Benefits arise when using ZFS filesystems on BSD/Solaris and Linux, when using compressed file systems to store logs.
Be sure to read about maximum message length limits and stuff, so you do not get into surprises.
Hope that helps
Regards
Peasant.