I may be wrong but I believe this to be a bug in clang and GCC.

Consider the code below. Both compilers complain (correctly) that the constructor A(int) is private in class D. Note that, as A is a virtual base class of D, A has to be initialized in the mem-initializer of class D, the most derived class, according to §12.6.2/7 in C++11. See live example.

class A {
public:
    A(int i) : x(i) { }
    A() : x(1) {}
    int x;
};

class B : private virtual A {
protected:
    B(int i) : A(i) { } };

class C : public B, private virtual A {
protected:
    C(int i) : A(i), B(i) { }
};

class D : public C {
public:
    D() : A(1), C(3) { }
};

int main() {
    D d;
}

But both compilers don't bother with the fact that the default constructor for class A is also private in D, i.e., both compile and execute the code normally, if we define the constructor for D as follows:

D() : C(3) {}

And this is wrong, as far as I can tell.

Note that both compilers fail to compile (correctly) if we define:

D() : A(), C(3) {}