You can edit customer chat room a department. 

For example, if a customer has no department, all staffs have access to the chat room message. Now you can assign/edit customer chat room to a specific department.

This can be done by double clicking the chat room title. 

Chat room members dialog shall open.

Scroll down the dialog and you can see "Add department" button in the dialog.

The screen also displays existing departments info (if any). You can remove the department as well.

Note a customer can have no department. But once you add it to a department, there has to be at least one department linking to the customer.