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To create a private and specialized custom copy constructor in C++, follow these steps:

1. Declare the class with a private constructor and copy-constructor.

2. Declare the copy-constructor as private.

3. Overload the new operator with a private function that creates a new instance of the class using the copy-constructor.

4. Define a friend class or function that is allowed to call the private copy constructor.

Here's an example:

class Foo {
private:
   Foo() {}
   Foo(const Foo& other) {} // Private copy constructor
public:
   static Foo createCopy(const Foo& other) { // Private overload of new operator
      return Foo(other);
   }
   friend class Bar; // Friend class that can call the private copy constructor
};

class Bar {
public:
   void doSomething(const Foo& foo) {
      Foo newFoo = Foo::createCopy(foo); // Calls private copy constructor
      // Do something with newFoo
   }
};

In this example, the Foo class has a private constructor and copy constructor. The createCopy function overloads the new operator and creates a new instance of Foo using the copy constructor. The Bar class is a friend of Foo and can call the private copy constructor through the createCopy function.