Is it a necessary trade-off that using smart pointers requires my class to be publicly instantiable?

I have a base product class with a few private members and a public getter that derived classes inherit. I would like to disqualify instantiation, since the class is intended for use with an abstract factory. I thought protected con/destructors might work, however, this breaks my smart pointers. Friending seems like a useful disaster. Is there a well-known solution to this, or should I resign myself to the fact that any client who has the factory injected must also know enough to instantiate the base product?

class Product
{
private:
    char type_name;
    char size_name;
public:
    Product(char, char);
    virtual ~Product() {}
    void Print();
};