I think there will be Bitcoin banks. They’ll work like banks do today, but they can operate globally and with lower costs because of the nature of Bitcoin.
The user can verify that the bank has enough reserves by examining the block chain. The bank publishes addresses it holds, and users can verify the balances. It’s a vast improvement over the current system where you just have to trust the bank’s word.
Yes, I do not believe that users will run full network nodes. The network will be maintained by specialists.