By putting your press release on your site first, you’re doing several things. You are keeping your site active with new content, which Google likes. You’re also establishing your site as the “authority” site for that story online, as it was published there first. Even if another site publishes your press release, Google will consider your site the originating site for the story and rank you higher. Also, you get some SEO influence from any relevant keywords in your press release.
Remember, a press release is content, and Google loves content.