This is not streaming, this is just torrenting and it includes uploading. The only difference between this and other torrent clients is that it downloads torrent files sequentially by default and opens the file as soon as you have enough data to start watching. From the FAQ:
Popcorn Time works using torrents, fair enough. Am I seeding while watching a movie?
Indeed, you are. You're going to be uploading bits and bits of the movie for as long as you're watching it on Popcorn Time.