Hi msta,
1. If "Remote Monitoring plug-in" (RMS) doesn't monitor is down, then an alert "HeartBeat" should appear in Event Console for this RMS machine.
2. Have you made sure that this remote RMS server is really monitoring computers? You can check this using "Real Time Performance" page to see whether it returns metric data for apprpropriate monitored resource.
3. If you want to monitor ICMP on "RMS" machine, then you just need to add these RMS machines in "Resource Target" of agentless Monitor policy and refresh policy on RMS machines, after that RMS will start monitoring itself.
--// Don't forget if you will start monitor RMS itself, then it will consume 1 license node for "Monitor Packs for Servers" product.
Thanks,
IP.