Why your ticketing system should not be accessible to users
Here's a thesis:
The problem I see with making ticketing systems accessible to users is that you periodically need to have a genuinely private conversation among the sysadmins in response to user requests. I'm not sure how often this happens in typical organizations; ours is somewhat atypical in how we structure support, and possibly as a result it seems to happen fairly often here.
If your ticketing system is (internally) public, then you can't have these conversations in the ticket; you have to find some other place for them. The best option is probably to have them in email that is not copied to the ticketing system, but even that means that your ticketing system is no longer the single place where the entire issue is kept track of, which raises the question of what it's for.
(Per OptionalTicketing, I don't think that it should be the official place that users interact with the sysadmins.)
Arguably your ticketing system can still be your tracking mechanism, but I think that it's going to be a relatively weak one; at best it can remind you of what's still in progress (and how important or urgent it is), since you may have to go to your email to find details. (You probably don't want to even mark tickets as 'we're having a private conversation', so you will also have to remember that you need to check email for the full status for particular tickets.)
The other problem of starting to use email for some conversations about tickets is that it is easy for such a conversation to wind up covering things that probably should be in the ticketing system so that the user can see them. In turn, this is going to frustrate either or both of you and your users (you as you have to copy information into the ticketing system; your users as the ticketing system no longer really reflects reality).