One of my Departments asked for this as well, I havn't looked into it as of yet. But am interested as well. We do wish that when the tickets are made public the identifiable information is striped out AND during ticket creation the submitter can choose to let staff see it only.
Our customized version of osTicket is used as a community support system to handle code enforcement, road repair, public works matter, tourism questions, etc. So there might be a time when the user doesn't want their neighbor knowing who ratted them out.