I think the author should be the actual author. It seems fair. That said, I haven't always read the stories before I list them, and I'm generally just grabbing and going in a big marathon posting session. This month, for instance, I was compiling over ninety of them at once. It takes over an hour to collect, tabulate, and insert that many into a poll, and then parse out the stories eligible for the Rookie Author award. So in general, I'm C&P-ing blindly.
Shall I assume one or more stories this month is someone else's? If they're labeled as such, please just point them out and I shall adjust the poll. If they're not labeled, let me know how to confirm authorship.
That said again, I wouldn't worry too much. There's nothing more at stake than some kudos.
As for porting your own stories from other sites, I see nothing wrong with that at all. As the author, you have total rights to re-use them as you like. For instance, though I consider RU to be my home site, I do post a unique story on another site for a contest there, and then re-post it here two months later once that contest has gone through, at which point it becomes eligible for our own contest. This month has a lot of reposts in the selection, but they're all new here (as far as I know) so I see no problem.
And finally, and I think this has been mentioned, any board member can request that their story not be included for any reason or no reason at all. No one's done this, but it's fine.