Theoretically they could already send this via S/MIME encrypted Mail (KIM) to the family doctor, but most Hospitals haven't rolled out this service yet.

They just started installing Card Readers for the Doctor Identity Cards, so they can issue electronic prescriptions

For communication with Patients some Hospitals have Web Portals/Apps for getting/sending information.

That's pretty interesting. We have electronic prescriptions too (though it goes straight to the pharmacy however - we don't see it).

As far as I know, I don't think the hospital portal has ever been used for communication like that. An email seems more "obvious" perhaps to the docs, and that's what they use most of the time.