Activision announced on Twitter that the beta is now live on both PS4 and Xbox One. Those on Xbox One have to preorder the full game to gain access, but the PS4 version of this beta is (unlike the first) open to everyone. The beta is–at least for now–scheduled to run until October 24.
The first beta got off to a rough start, but Activision and developer Infinity Ward were able to iron those out. New content was introduced over the weekend, including additional game modes–you can read all about the maps and modes here. Players also found a glitch that allowed them to fly, though Infinity Ward said it was aware of that and would stomp it out.