Hi Brad,
I'd actually take the opposite position of Alan. I would distribute the press release first and then post it to your site a bit later. I see press releases ability to build external links as one of their biggest values. If there is a concern for duplication, I'd rather have the external postings of the release 'count' in the eyes of google as opposed to the internal.
I wouldn't worry too much about it either way though. Press releases are posted and reposted all over the web, and that's just the way it goes.
Your press release should be linking to important content on your site. It's unlikely that the brief content on the PR would be a significant source of organic traffic for your own site, opposed to the content the release is actually writing about and linking to.
The only time I would do it the other way around, would be if I was creating a PR for a company that is popular and newsworthy no matter what. If SEOmoz had a press release, it would make sense to post it on their own site first because people within the industry are going to find it and link to it, therefore creating many more external links to their actual domain.
Long story short, in my opinion, new/small website should submit first. Big/popular website should post on their own site first.