Isn’t it a waste of time speaking to two different operators for every emergency?

No, the decade of experience acquired by the Single European Emergency Number 112 and performance assessments have shown that speaking firstly with the Single European Emergency Number operator and then with the operator of the relevant emergency service (Carabinieri, Police, Fire Service, Emergency Medical Services, Coastguard) is not a waste of time because it guarantees faster localisation and the coordination of all emergency services, speeding up response times.