If I'm streaming video from my phone to Periscope's servers: is my phone being a server and listening on a random port or I'm just sending my stream?
For example: VLC makes your device a streaming server, does Periscope do the same?
As far as I'm aware, your phone is just uploading a stream to their (Periscope's) service, from which it's being distributed to anyone who's watching it.
This is the way such "live-streaming" usually works. A user's upload bandwidth isn't usually sufficient for streaming to multiple clients, but their servers are.