The answer is mixed. While the iTunes app store can be signed into with a separate Apple ID, there will be some data (naturally) stored on the second account you use. The device will not be added to their My Apple Devices, though it will become an Authorized Device in iTunes Store. As such, some level of data will be held in the Apple Servers to facilitate this transaction. AFAIK this data is not user-accessible.
However, for data backups, Photos, Find my iPhone, My Apple Devices, etc. that would require the user to login via iCloud on the phone and not just the Apple Store. From there, whatever level of data you set to sync would be replicated up. Please note, you CAN be signed into a separate iCloud and iTunes account, as well as various other services (such as separate iCloud and Messages accounts).