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Asking how many hands we need for such a study is the wrong question. First we should ask what kind of checks we would like to do on the data.

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That's a good point. What are you wanting to check?

If you want to know if the cards come out of the deck in the right "odds" as they are meant to that would be easy enough to do sample size wise.
However if you wanted to make sure certain players weren't "Rigged" then that would almost be impossible because you would never get a large enough sample size (unless they were silly enough to upload their HHs)

If someone was really worried about a person/group having all this "important" data then you'd just need to obscure the information before sending it The players names aren't important. So someone can write a really simple program that just replaces player names with UTG UTG+1 SB BB etc. People then run their hand histories through that parser. Then upload it to the central database.. Really does player names matter?
All you want is one HH file of every hand ever played that's gone to showdown right? people seem to usually complain about the showdown riggedness when there hand doesn't hold up. All of this information is available when observed.

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For certain types of analysis, like determining if there is such a thing as boomswitch / doomswitch or rigging to help fish survive etc, you would need to each player uniquely identified...because you'd be trying to determine how one player runs versus the population....or how a group of players with similar attributes (ie high vpip/high wtsd) perform versus the population. But the unique identifier doesnt have to be the screenname.