No, you do not need to be logged in order to receive alerts. However the CMS, or Central Management Server, must have the Insight Manager server service running. The CMS is where you actually install SIM, and is where you tell SNMP on your managed servers to send traps to.
What you do is once you have the MIBs compiled into SIM, you set up an "Automatic Event Handler". An Event Handler basically tells SIM to do something when it receives a particular type of trap, a trap from a particular machine or group of machines, or some combination of the two. You can have it send email, page, run a script, etc...