To be precise, we don't necessarily exchange properties, we exchange our respective claims of ownership to what we have. 

We might for example write a check, which represents our agreement to transfer possession of a certain amount of money that we have on deposit at the bank, in exchange for the title of ownership to a painting that resides safely in the vault of the auction house.

Often we do make an actual physical exchange on the spot.  We might purchase for example a basket of groceries at the market, transferring into the hands of the grocer the actual money involved in the purchase.

Of course, to be more precise, even money as we know today it is little more than a representation of a property of value that resides elsewhere.  Unless the money is itself made of a valuable substance (such as silver, gold, or diamonds), it generally has value only in what we can exchange it for.

So the exchange is completed and we believe that we exchanged items of property.  What we really did was simply to transfer our respective claims to the property.