@Adam: That's possible today, though not exactly as you described.

We have the concept of "shared" organizations in Zendesk. When an Organization is marked as shared, it means any member of that Organization can view the tickets submitted by themselves and their peers. You can go a step further and also allow them to collaborate on each other's tickets, but that's a further option of Shared organizations, and not a default.

Member of shared organizations can go to web portal, clicking on the name of their organization in the top menu (or clicking "Organization Requests", I'm working from memory here!) and then they'll see a subscribe link location near the top right of the page. Each time someone submits a ticket or a comment, anyone subscribed gets an email notification.