You can absolutely post the fic to your IJ, but we'd prefer if you also posted it here, so we can keep everything in one place (though you're more than welcome to also include a link to your personal journal).
If you finish your first fic before the community opens for posting, and you want to choose a second prompt, just let us know via comments and we'll give it to you. No need to wait until after the first one is posted. We trust you :)