This happened with what eventually became Star Trek: Generations. Ron Moore & Brannon Braga wrote the screenplay that was eventually produced, while Maurice Hurley wrote an entirely different screenplay that was eventually rejected.

Producer Rick Berman invited fellow Trek producer/co-creator of DS9 and Voyager Michael Piller to submit a screenplay as well, but Piller disliked the idea of this competition and bowed out.

I doubt that's the only other instance and you'll probably find this more common with older franchises. I know Freddy vs. Jason went through several iterations, but I'm not sure if any of them were concurrently written.

A long time ago one of the friday the 13th movies, not freddy/jason had competing writers. More than two, one of them was Peter Jackson and his partner Fran Walsh, so it must of been a long time ago. I think between Frighteners and Heavenly Creatures.