“the client must at least ask the server how much money it has on its addresses”, correct, but in bitcoin system, the btc addresses are not private data. Every transaction a user made, is seen by all the other peer, including involved addresses and amount of coins and where from/to. Check
http://www.blockexplorer.com to see more details.
So, it means that the "private data" you quoted is entirely known by every one in bitcoin network not only the server.
I know how it works, and no, my private data is not entirely known by everyone because you can't know a priori which are all my address. You can't even know whether two particular addresses belong to the same person or not.
In a client server solution like yours, the server can link together all the address of one client wallet. Okay, if it doesn't ask anything more, it will only know the IP of the owner, not his identity. But it adds some risk anyway.
I know it's not a major problem, mainly because people should not put all their wealth in a smartphone, but I'd suggest you not to keep any logs in your server that could link together different addresses, just for precaution. If you keep such logs and they leak, that could be bad, since anyone that has ever done one single transaction with one of your clients could then know every other transaction that this client ever did using your app.