If you are running unix, linux, or any of the BSD variants, your logging facility is syslog. In it you can log to a syslog daemon on another host.
Mac is mostly unix under the hood, but apple has hacked it a lot. It may or may not work.
Windows? havn't a sniff.
There's another program, netcat, that in effect constructs a network pipe. Log to a file. Point netcat with one end watching the file, and the other end sending data to a host.
Note that for timestamps to make sense, you need to have everyone agreeing what time it is. If you don't already, run ntpd on all your machines, and configure one to sync to some time standard, and broadcast time to your local network.
You also may need to do some data massaging to get finer time granularity. Most timestamps are only to the second. If you are logging multiple services from multiple hosts, you want a strict time sequence.