I just looked at my privileges, and I'm currently 86% of the way to having permission to retag questions. When I retag my first question, I'll get the Organizer bronze badge. Except, if you look at my profile, I already have that badge, because I've already retagged a question: I added the flat-file tag to this one
It seems that when you click on edit, you can add and remove tags regardless of whether you have the retag privilege. If you're not supposed to be able to retag questions until you have that privilege, then tags should probably not be shown when editing a question.
Edit: Signing up for meta.stackoverflow.com gave me an additional 100 reputation on the site, so I now have that privilege legitimately, but at the time I performed the linked retag I did not have the privilege to do so. So looking at my account now will not reflect the state it was in when I performed the retag. Just so people don't get confused or think I'm trying to mislead.