Okay, great questions. There are two different types of storage. One is the Paper Vault, where you store your Private Keys yourself. We never see your Private Keys, never have access to them, and your Private Keys are never seen or sent over the internet. It is the safest way to store your BTCs.
 
Then there is Active Storage, which is less secure but more convenient than Paper Vaults. We advise you keep as few BTCs there as needed, and the lionshare in your secure Paper Vault (much like you would typically keep smaller amounts of spending cash in your back pocket, and keep your life savings in a bank). When you want to spend your BTCs out of the Paper Vault, you can either spend them directly out of your Paper Vault (which requires going through the network)- or if you want to transfer BTCs to another Bank member for free and instantly, you can log in, enter your Private Key on the clientside javascript page we provide (so your Private Key is never revealed online, and your BTCs are never made vulnerable- only the transaction string is sent) and you can send the amount you need to your Active Storage for spending. In Active Storage, your Private Keys are kept on our server like Gox does it, which is how you can send to other Members free and instantly. You can always just transfer BTCs into Active Storage when you're going to immediately spend them, significantly decreasing any changes of anyone hacking our server, and therefore gaining access to your BTCs. Each type of storage has it's own benefits and disadvantages, but the tradeoff of storing your BTCs in Paper Vaults is pretty small- it's just the time it takes you enter your Private Key.