To the extent we are aware, no privacy issues have been raised in relation to purchasing apps.
In order to purchase an app or even to download a free one, the user must be logged into Apple using an Apple ID and password, which is linked to a credit card. So while the user does not sign up each time an app is purchased and downloaded–only enters the password–the information is stores at Apple, not in the iPad itself.
Through their agreement with Apple, developers are prohibited from discussing how the system works or what information they receive. Granted this is not much of a protection, but it is in place. We have yet to see apps specifically survey users for personal information. Having said that, an app can make a user log in to another system to work and that second system could make users provide personal information. There is nothing in the Apple iPad system to suggest any more or less risk than any other computer-related system. Like any other computer system, the best way to manage the risk with these devices is to be involved with what users of the device are doing–especially children. One of the simplest ways to manage this is to prevent the user from downloading or buying apps on the iPad altogether.