I thought I might promote the site a bit locally (by putting up flyers by the elevators, etc.). Is that OK to do during the beta, or should I wait for the site to launch before promoting it to people who are not already familiar with the StackExchange engine?
It's perfectly fine to promote the site during the beta, it's probably even the most important time for promotion. We need to attract more users to get the site ready for graduation.
Right now, only existing users of the SE network have a significant chance to stumble upon this site. We don't get a lot of traffic from search engines in the early days.
Probably the best way to promote the site is to directly invite other experts that you think might be interested. Flyers in appropriate areas like universities are of course also a good way to attract more users.