Who gets to download the data?
From what I have heard (and this is just what I have heard from usually reliable sources), the Malaysian Authorities (who have primary legal right to the data) have agreed that it will be accessed by the Australian Transport Safety Board, the (US) National Transport Safety Board, and a Chinese authority I can’t remember the name of. I suspect it will be primarily accessed by the ATSB simply for convenience (Australia’s facilities are closest), unless the recovery is complex in which case the NTSB are the best in the world for difficult data recovery, but the Malaysians would have the legal right to insist on doing it themselves.
I would think that only Boeing, or the manufacturer of the recorder, can safely access it. I doubt that it has a USB jack that you can connect with a cable to any desktop PC and it immediately starts playback. As a bare minimum, the outer protective container must be opened - and they are made to survive in fire, or as this case may be, at the bottom of the ocean. You don't want to do this with a sledgehammer or a plasma cutter. You need drawings, documents, and special tools. Boeing has them. I doubt that the customer airline has them as well. It's not something they need to maintain from the inside - it's just a board with Flash chips. If it passes self-test, it's good to go. If it doesn't, back to the manufacturer it goes.
Malasia gets first dibs.
But they may defer to NTSB, who is acknowledged to be the best at forensic retrieval.
Certainly NOT Boeing, and NOT Malaysia Airlines.