You would need to know the exact millisecond of the timestamp and usernames to access it. Our plans are to allow sharing of video/voice messages publicly if the user chooses. Feedback welcome...
https://twitter.com/ryanseys/status/355009243306405890
EDIT: Also, the timestamp is per-second (1rajmadugulaedwardallison12013-04-23T17:06:49Z.wav) not per millisecond, making bruteforce a rather elementary task when combined with twitter tweet information.
I did not need to know anything else. This is not good.
Since this uses Twitter, a good move may be to use Twitter's API to check if the correct user has been logged in (I don't know the Twitter API).
Would using direct messages work?
Not exactly caring about the users' privacy, or security.
[I have poor impulse control, sorry.]