If you don't have exclusive control of the private keys, you don't have the bitcoins.
Blockchain.info is a wallet. You have exclusive control over your private keys. blockchain.info does not have the ability to move bitcoins between blockchain.info wallets without using bitcoin transactions that must be transmitted over the bitcoin network and confirmed in the blockchain.
CoinBase is NOT a bitcoin wallet. CoinBase is an unregulated, unaudited, uninsured, bitcoin bank account. They could even be engaging in fractional reserve, and there is no way for you to know. You make a donation to CoinBase, and in exchange they give you a promise that they will send an equivalent amount of
their bitcoins wherever you ask them to whenever you ask them to. As long as they are willing and able to honor that promise, everything works ok. If for some reason they become unwilling or unable to honor that promise, there isn't much you can do. The bitcoins are no longer yours, since they own all the bitcoins that they display in "accounts" to their users.
An "account" is just an entry in their personal database indicating how many bitcoins a particular user can make requests about. To "send" a microtransaction from one CoinBase account to another CoinBase account, they don't have to actually move any of their bitcoins at all. They just reduce the value in the database entry on their system for the "sending" user, and increase the value by the same amount in the database entry on their system for the "receiving" user.
