The fire service has several programs available for incident notification via texts, which could be considered analogous to crew calling.
One feature several of them have is the ability to respond, with several options. When a call comes in I can respond that I'm headed for the station, the scene, I'll be delayed, or maybe I can't make it at all. My phone number is registered in the system, and there's a default response (mine is going to the station), so when I call, the system recognizes the number as me.
While responding to the notification is usually done by phone, my smart phone has an app that gives me a net based response capability, and even will show who else has called in.
The basic texts tell me what the call is, and where. Some will even port to a map program and help me find my way to the call (not usually necessary).
It wouldn't be a reach to come up with a similar program for crew calling, although it does require that the callees have a textable phone. I would imagine that a hybrid system could be constructed that uses both texts and landlines for notification.
One of our firefighters doesn't have a cell phone, but does carry a voice pager (our primary notification method). He still calls in from his home phone when appropriate.
The system we use has the ability to send text messages to any or all of the folks in the system (for our department), so if applied to crew calling, the caller could notify just the appropriate crew members, or send out status messages (ie, train lineup for the next X hours).
The system also has a web presence, so I can log in and see what's up, as well as schedule myself, including those times when I won't be available to answer calls.
Or they may already be using something like that.