[21.4.1.4] does say "Every object of type basic_string<charT, traits, Allocator> shall use an object of type Allocator to allocate and free storage for the contained charT objects as needed", but that "as needed" most definitely leaves the door open for a COW implementation.
The only other part that I can see that may preclude a COW implementation is the postconditions specified for copy operations [21.4.2], which says data() returns a pointer which "points at the first element of an allocated copy of the array whose first element is pointed at by str.data()". Again though, "allocated copy" doesn't necessarily mean "a copy I just allocated". When I go and get a copy of a book I don't literally go and copy it.
In fact the standard specifies that the move constructor leave the source value "in a valid state with an unspecified value"... which again suggests you could use COW and have the source argument return the same value it had before you moved from it.
In the latest C++14 draft though (N3690), you're right, it is explicitly prohibited because "Invalidation is subtly different with reference-counted strings".