Large companies tend to do their logging/monitoring centrally. They have thousands of devices (computers, disk arrays, switches, routers, tape robots, etc.) and only a handful of staff to monitor. Centralization is a necessity. There are commercial products like HP Openview and Tivoli.

Central logging as another benefit: central logging implies remote logging. Remote logging means that if a machine goes haywire (or, in a hostile environment, get corrupted), it's less likely logs get wiped out.

In many places I've worked, be it as an employee or a contractor, some sort of central logging was done. From simple things as FTP'ing local logs in a nightly batch job, to thousands of machines monitored/logged with Tivoli, displaying the current status of the environment on a monitor 3 metres wide.

Personally, I'd go for central logs. If only because that means I know where to go digging for possible log file entries. But then, I look at the problem more from a sysadmin angle than a programmers' or end-users'.