From the demo, it seems like the recording is uploaded on their website by default. It is a bit concerning.
> If the filename argument is omitted then (after asking for confirmation) the resulting asciicast is uploaded to asciinema-server (by default to asciinema.org), where it can be watched and shared.
If the filename argument is given then the resulting recording (called asciicast) is saved to a local file. It can later be replayed with asciinema play <filename> and/or uploaded to asciinema server with asciinema upload <filename>.
You can also roll your own tool, the format is reasonably well documented and simple.