If you have an iPhone and a Mac computer, simply connect the iPhone via the Lightning cable to the computer and startup Quicktime Player.
Go up to File -> New Movie Recording and your camera will likely boot up with a Play-FF-RWD-Record like button at the bottom.
Click on the prompt there and you’ll see an option to change input devices.
Select iPhone for both Audio and Video, and you now have the capability to record whatever is on your phone directly to your computer.
Open up the Periscope App. Dig into your profile Recent Broadcast and select a recent broadcast. Click play on your phone, then click record on your Mac, and let the stream come through, recording not only the video but the comments as well.
When all done save the file and there you have it–a nice recording of your Periscope video with the comments.
Now go make some great scopes!