Imagine this picture...
You have a full client on your Iphone running in the background, and then there is a power failure. You head down to the corner store, and find that the shopkeeper has put everything in the cooler on sale half price, cash or bitcoins only. Your cellphone client connects with the shopkeeper's cell phone client over ad-hoc bluetooth. Signs the transfer accouncement (there are no actual cryptocoins in your wallet, they exist only as a series of entries into an encrypted ledger we call the blockchain, more like writing a check than actual coins) over to the shopkeeper's address. Shopkeeper's client can then (but does not have to) check his copy of the blockchain to verify that you actually owned said bitcoins at the time of his last blockchain update. If it's good locally, he can assume that you are not trying to cheat him and accept the trade and you leave with your half priced milk. This does not protect the shopkeeper from an intentional double spend, but you still had to have honestly owned the coins at one time in order to do this. If you did not own the coins at the time the power went out, his client would have rejected the transfer.
There was another point here that I intended to make.
As with the example above, most cash transactions are not actually anonymous, but psuedo-anonymous. For example, one party can be well established (the shopkeeper) while the other is mostly anonymous, (the bargain hunter in the above example) but even the buyer is not truely anonymous. He has been in the shop many times before, and even if the shopkeeper doesn't know his name, he has seen his face enough times to recognize it. So there is some trust that, since he is a local or regular customer, that he is not some pro trying to sting him with an opprotunistic scam.
Another example of a psuedo-anonymous transaction is the kind that both parties are anonymous to the world, but not to each other. This is actually how *most* cash transactions work in the real world; but online it could go like this...
You want to get something, let's say, unusual. You find this guy on, say, Tor with a hidden website selling your wanted commodity. He is known to you only as, "pothead420" on the website. You have no other way to contact him, and no means of finding out who is actually is. You don't trust this guy, but someone is going to have to take a leap of faith, and that someone is likely going to be you. So at first, you order small. After a few succesful trades, your trust grows that "pothead420" is an honest dealer; so your orders grow larger. He could screw you over at any time without recourse, but then he would be cutting off a regular customer, so he has an incentive to continue to treat you properly.
A third kind of pseudo-anonymous trust relationship could be found on this very forum that depends on reputation. Most of us do not use our real names, but some of us do. We assume that the member who uses the name of the bitcoin programmer is the actual programmer, and have some evidence to support this, but we cannot know for sure. But, as far as this forum is concerned, he has a reputation. So if you were to do direct business with him, and screw him, his word that you are not trustworthly would harm any business that you desired to conduct in the future, solely because he has a longer reputation than any newcomer.
All of these examples involve some kind of two party trust, but not one requires both parties trust a third. Not even the verification of the blockchain.