I'll admit, I haven't applied for access through either one, but neither have I seen any papers cite access through those venues—and I read quite a few NLP + Twitter papers.
This article is just talking about Twitter Data Grants for which 6 universities were decided as winners [0]. You won't see papers through these grants as yet because well, the winners were announced about 40 days back!
In December, 2010, Twitter named a Colorado-based company, Gnip, as the delivery agent for moving data to the Library.
Shortly thereafter, the Library and Gnip began to agree on specifications and processes for the transfer of files - "current" tweets - on an ongoing basis.
In February 2011, transfer of "current" tweets was initiated and began with tweets from December 2010.
On February 28, 2012, the Library received the 2006-2010 archive through Gnip in three compressed files totaling 2.3 terabytes. When uncompressed the files total 20 terabytes. The files contained approximately 21 billion tweets, each with more than 50 accompanying metadata fields, such as place and description.
As of December 1, 2012,the Library has received more than 150 billion additional tweets and corresponding metadata, for a total including the 2006-2010 archive of approximately 170 billion tweets totaling 133.2 terabytes for two compressed copies."
I find the quantities hilarious.
But since they haven't been able to cope with providing access yet I get pessimistic about their prospects of doing so at all any time soon.
Can we do something to help them?
I've been thinking maybe GPU-accelerated databases like MapD, could mitigate the cost issue for them, but I'm pretty sure that doesn't go all the way to solving the problem...
[0] https://blog.twitter.com/2014/twitter-datagrants-selections