The Republican governor of Mississippi, of all people, explains convincingly why 'herd immunity' is a pipe dream. Well worth reading.
If you watch Toms Hanks interview on Stephen Colbert, he had the virus, and his antibodies to the virus have gradually dropped over time. So it doesn't appear people have long term immunity. It will be interesting in the coming months how many people do get reinfected with it. His experience with the virus didn't sound too pleasant either.
read this article about why the t cells might show more about immunity to the virus