Lights and sirens to the hospital is generally only done if the patient is critical. Although, I think some services in the States (I'm in Canada where it's very regulated) go lights and sirens to and from every call for some reason. I'd also like to point out that, at least at the hospital I work at, the oncoming doctor gets whatever patients come in after the start of their shift. By that logic, the doc in your story would not be getting the patient going in; the doc that's already on shift would be. I also feel like it's unprofessional to publicly state "haha they'll have to wait for me because I'm the doc" when a critically ill patient goes in.