This doesn't compile:

#include <memory>
#include <sstream>
#include <cassert>

template<typename T>
struct Alloc : std::allocator<T>
{
  template<typename U> struct rebind { using other = Alloc<U>; };
  Alloc(int id) : id(id) { }
  template<typename U> Alloc(const Alloc<U>& a) : id(a.id) { }
  int id;
};

using string = std::basic_string<char, std::char_traits<char>, Alloc<char>>;
using stringbuf = std::basic_stringbuf<char, std::char_traits<char>, Alloc<char>>;

int main()
{
  string s(Alloc<char>(1));
  stringbuf b(s);
  assert( b.str().get_allocator() == s.get_allocator() );
}

The basic_stringbuf constructor default-initializes its basic_string member, which default-initializes its allocator.