Bungie yesterday released a free trial version of Destiny for all platforms, allowing players who don't own the game to experience a sampling of gameplay and character progression.
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Called the Destiny Trial on PlayStation 4 and Xbox One and the Destiny Demo on PlayStation 3 and Xbox 360, the free version is now live for a rather large download. It's significantly larger on the new consoles--20GB on PS4 and Xbox One as opposed to 6GB on the last gen systems--but those are actually the same sizes as the full retail versions.
The identical file sizes compared to the real game is likely due to the nature of the trials: it sounds like Bungie took some of the full game and capped it off, rather than creating a separate demo. The nature of Destiny requires integration into the real game world and matchmaking to operate, so a standalone demo wouldn't really give the proper experience.
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What's more is that your demo progress can carry over to the full game if you decide to buy, so long as you stay in the same system family. Character and profile data is stored on Destiny's servers, so PS3 files can transfer to PS4, and Xbox 360 games will convert to Xbox One (but no PlayStation to Xbox). As long as you have enough hard drive space, the demo is available for download right now.