Q:

Why does it not work to overload "operator <<" on std::string?

#include <string>
#include <type_traits>

using namespace std;

template
<
    typename CharT,
    template<typename> class Traits,
    template<typename> class Allocator,
    typename RightT,
    typename StringT = basic_string
    <
    CharT,
    Traits<CharT>,
    Allocator<CharT>
    >
>
enable_if_t
<
    is_constructible<StringT, RightT>::value,
    StringT&
>
operator <<(StringT& lhs, const RightT& rhs)
{
    return lhs.append(rhs);
}

int main()
{
    string s1, s2;
    s1 << s2; // compilation error!

    return 0;
}

My compiler is VS 2015 Update 3. The compilation error message is:

error : invalid operands to binary expression ('string' (aka
  'basic_string, allocator >') and
  'string')

Why does it not work as expected?

A:

Did the compiler give you any more info than that single line?  Something like:
26:1: note: template argument deduction/substitution failed:
35:11: note: couldn't deduce template parameter 'CharT'

If you replace 
operator <<(StringT & lhs, const RightT& rhs)

with
operator <<(basic_string<CharT, Traits<CharT>, Allocator<CharT>>& lhs, const RightT& rhs)

it compiles.
Basically you are putting the cart before the horse.  You can use template args to form a default template arg (StringT = ...) if you already know the template args.  You can't use the default to determine the args.
If you want to support both basic_string and other/custom strings, you may need to write two specializations or something.
Or realize what your template requirements really are - you don't require Constructible, you require "Appendable", so SFINAE on that, and ignore whether it is a basic_string or MyCustomString - it doesn't matter; the only thing that matters is whether lhs.append(rhs) works (and, uh, maybe figure out its return type as well...)

