Ron, there are several benefits in using a remote log shipping monitor. They include
1. Centralized monitoring and alerting solution. If you have several instances participating in log shipping throughout your enterprise this would allow you to view the overall health of those instances (provided you setup a remote log shipping monitor :-D).
2. The transaction log shipping status report on the monitor displays the status of backup, copy and restores for the instances using the monitor (instead of having to look at each primary and secondary instances). Those values listed in red have exceeded their set thresholds and action is needed.
3. If either the primary or secondary instance goes offline, or SQL Server Agent is stopped, you'll still receive alerts from the remote monitor. For example, if the primary went offline you'd no longer receive alerts when your backup threshold was exceeded.
For more information take a look at the link below. Thanks for reading!
MCSE: Data Platform, MCP, MCTS, MCITP - Database Administrator & Database Developer