class declaration over different files

There is a way to get something quite similar: private inheritance.

// private.hpp
class test_details {
  protected:
    int a;
};

// public.hpp

#include "private.hpp"

class test : private test_details {
  public:
    int geta() const { return a; }
    void seta(int i) { a = i; }
};

Note that you will still need to (indirectly) include the private header in any module that uses the public class, so you're not really hiding anything this way.

Not like that, but the pimpl idiom (or opaque pointer, or Chesshire cat) can help you achieve similar functionality - you can provide a public interface where all implementation details are hidden in an implementation member.

C++ doesn't support partial classes.

Also, note that what you have there are class definitions, not declarations. C++ mandates that if multiple definitions of a class are available, they must be identical, otherwise it's undefined behavior.