In a smaller MSP it is usually a function rather than a place: the same technicians who take support calls also work the alert queue. In a larger one, or where it is bought as a service, it is a separate team whose entire job is the monitoring queue, often around the clock.
The distinction worth holding is between NOC and SOC. A NOC is concerned with availability and performance — is it up, is it slow, is a disk filling. A SOC is concerned with security — is this behaviour hostile. The skills, the tooling and the urgency are different, and a team told to do both usually does the one with the louder alerts.
Outsourcing a NOC is common and reasonable, because monitoring benefits from scale and someone awake. What to establish before signing is what the provider does when they find something: whether they remediate, or whether they raise a ticket for your team and consider their obligation met.