If you have the XDK you can connect to an XLSP server which stores this stuff, this is how racing games do ghost laps of leaderboard data most of the time for example. There is a whole bunch of TCR's you have to comply with though if you want to release the game.
It's easily doable on Windows. For XBox, why not when the game is beat, upload just the relevant data to your server, and generate a unique key for that save (example: a32bd45u), that the user can type into the sequel to download the imported save file from your server? I've never messed with XBox development before, so I don't know if Microsoft blocks network access for downloadable XBox games.
Also side note you have to manage your own XLSP server, this is not something MS will host for you.