Which is a problem that still affects me too. I only mentioned the broadcast aspect because:
a) It shows that Social is connected to the Twitter account and can post.
b) It was something that had been mentioned as not working in other threads since the release of ver. 3.0.
c) It shows that the tweet sent through the broadcast is entered into the database on the WordPress end (although it hasn't recorded @ mentions or hashtags in the tweets db/posts).
I still haven't found a solution to the tweet downloading issue, but my best guess is that it either relates to the code that is supposed to do it or the cron execution that is suposed to trigger it.
Note: That's WordPress cron, not a creal cronjob.
Unfortunately I'm not a PHP coder, so my ability to analyse the source is limited. Python is more my thing, but that doesn't really help us here.