Bungie has confirmed just now in the IGN live stream that the Destiny beta on Xbox One will not be running at 1080p. The game itself at launch will reach 1080p. Bungie’s Community Manager said that ‘there’s always time for improvement.’
Bungie is aiming to make the game run at 1080p on both PS4 and Xbox One, and it seems that the beta will be at 1080p on PS4 only. Bungie says they want to make sure all platforms can reach 1080 by launch. Bungie did say at E3 that the game will be running at 1080p at 30FPS on both PS4 and Xbox One at launch.
IGN has stated that next week, before the Xbox beta starts, they will be debuting the first ever Xbox One footage of the game. The Xbox One footage of the game will be running at 1080p, but the beta will not be running at the footage quality.