The Coinbase iOS app has launched
blog.coinbase.com
blog.coinbase.com
Blockchain.info web interface I believe works on the iPhone, albeit It's been a while since I've been there. If you really want to embrace Bitcoin, build from source https://github.com/bitcoin/bitcoin
The advantage of bitcoin is being able to have that option, not that it's always necessarily better to store bitcoin on your own.
If you are new to Bitcoin apps, you may want to check out Gliph. [1]
The app lets you view your Coinbase, Blockchain.info or BIPS bitcoin wallet and send Bitcoin to people easily.
[1] https://gli.ph/ (I work on Gliph.)
I wonder if Apple will start allowing entirely phone-based wallet apps now...
The top screenshot seems to show such an audit trail. I hope it is stored on the device (never on Coinbase's servers) and only temporarily.
You can think of the Coinbase app as a contact book that attaches a name and face to a bitcoin address. This is exclusive to the Coinbase servers. No one else but you can see the identities of the people you have transacted with, only the addresses and amounts.
However you (or anyone) can set up lots of accounts for free and unless you tell someone, people can't easily match up account to actual person.