Do they just call 911? Or is there some special unit that’s specifically trained around disasters and rescue?
Most countries have established some sort of Rescue Coordination Centre responsible for search and rescue operations in case an aircraft or ship goes missing. The RCC typically has direct access to various SAR resources such as helicopters. They can also call in local emergency teams (police, firefighting, ambulances etc.)
How exactly the RCC is notified of an incident can be different. It may be through a normal emergency telephone number (such as 911) or via some direct communication channel.
In the US, the ARTCC where the incident occurs acts as the focal point for search and rescue operations. They work directly with the US Air Force Rescue Coordination Center, local, state, and Federal law enforcement (if necessary), and the Civil Air Patrol during search operations.