It is unknown how many keys will be sent out and to whom. Last month, CCP told Fanfest attendees that they would be among the first to test the anticipated MMOFPS. The DUST 514 website states that the studio is sending out Fanfest keys first and keys to standard registered players "soon after."
To celebrate the release of the closed beta keys, CCP is holding a "special event" over the weekend. The details of this event are forthcoming.
[Update: CCP has confirmed that beta registration has begun today and that EVE players and Fanfest attendees will get first pick.]