The monitor thread group will run along with the other thread groups for your tests. If you set the monitor thread group to loop forever, you will need to stop the test manually to stop the monitor thread.
There are at least a coule of approaches to monitoring multiple servers. I just set up a different thread group for each server I wanted to monitor with a different HTTP request and different filename for the output script. Each thread group fires the request for 1 server and outputs the data to different files.