Warning for Xbox 360: HDD Only (Destiny)
The beta won't work from a flash drive -- you HAVE to have it on an HDD (which it doesn't tell you until you've fully downloaded it). So before downloading, move content from your HDD to your flash drive and then download the beta to your HDD.
What is it with Bungie games requiring an HDD? Every other game I've ever tried has worked perfectly fine from a flash drive.