I think there is a confusion here. The problem is not about headers. Headers don't do anything (they are just ways to include common bits of source text among several source-code files).
The problem, as much as there is one, is that class declarations in C++ have to define everything, public and private, that an instance needs to have in order to work. (The same is true of Java, but the way reference to externally-compiled classes works makes the use of anything like shared headers unnecessary.)
It is in the nature of common Object-Oriented Technologies (not just the C++ one) that someone needs to know the concrete class that is used and how to use its constructor to deliver an implementation, even if you are using only the public parts. The device in (3, below) hides it. The practice in (1, below) separates the concerns, whether you do (3) or not.
Use abstract classes that define only the public parts, mainly methods, and let the implementation class inherit from that abstract class. So, using the usual convention for headers, there is an abstract.hpp that is shared around. There is also an implementation.hpp that declares the inherited class and that is only passed around to the modules that implement methods of the implementation. The implementation.hpp file will #include "abstract.hpp" for use in the class declaration it makes, so that there is a single maintenance point for the declaration of the abstracted interface.
Now, if you want to enforce hiding of the implementation class declaration, you need to have some way of requesting construction of a concrete instance without possessing the specific, complete class declaration: you can't use new and you can't use local instances. (You can delete though.) Introduction of helper functions (including methods on other classes that deliver references to class instances) is the substitute.
Along with or as part of the header file that is used as the shared definition for the abstract class/interface, include function signatures for external helper functions. These function should be implemented in modules that are part of the specific class implementations (so they see the full class declaration and can exercise the constructor). The signature of the helper function is probably much like that of the constructor, but it returns an instance reference as a result (This constructor proxy can return a NULL pointer and it can even throw exceptions if you like that sort of thing). The helper function constructs a particular implementation instance and returns it cast as a reference to an instance of the abstract class.
Mission accomplished.
Oh, and recompilation and relinking should work the way you want, avoiding recompilation of calling modules when only the implementation changes (since the calling module no longer does any storage allocations for the implementations).