Only if bitstamp wanted to hurt their reputuation with no upside for them. "Hey potential customer, don't worry your personal info is completely safe, unless random people on the internet want it." 
That's the problem here, if the exchange controls all the customer information (their order and their money) and keep it as a secret, then they will become the single point of failure in the bitcoin ecosystem, just like what happened with MTGOX, lost hundreds of thousands of coins without anyone knowing what had happened (I strongly doubt that they really lost those coins, they can quietly stay in Marc's private wallet).
If the exchanges are transparent throughout the network like bitcoin, then such risk is much less. So far, only btc-china provided third party audit for their cold wallet and database, this is a way to prove that they do not manipulate the trade