I'm also puzzled how the emergency info is supposed to work.
Maybe it's restricted to US only, like several other features in Android?
I'm in Denmark, Europe, at Telenor - in case somebody knows what is going on.
The Emergency Information now works as expected on my phone after Android 7.1.2, the October 2017 security, and a carrier settings update.
There's now a button on the emergency dialler screen, to tap twice and reveal two panes with personal emergency info and the pre-selected contact(s).