Can the service providing the bitcoin wallet know how much value has the user loaded into his wallet?
All Bitcoin transactions are publicly available. This is fundamentally why Bitcoin can work as a truly decentralized currency. Bitcoin is different from the traditional banking system specifically because anybody can see and validate everyone's transactions. If only a few could access the information about people's balance, then it wouldn't be fully decentralized, it would be just like traditional banking.

So yes, you can calculate the balance of any Bitcoin address in the network.