What's the scope of window handles?

Someone stated (though I've yet to find any supporting MSDN documents) that window handles are global to all processes belonging to the same window station. Yet, if you somehow obtain a handle to a window belonging to another desktop, and try to send messages to it, SendMessage may fail with ERROR_INVALID_WINDOW_HANDLE. Why?

If window handles are truly global, then, say, thread 'a' on desktop 'A' should be able to use handles from thread 'b' on desktop 'B', assuming that both desktops belong to the same window station, right? It seems that some magic needs to happen for the handle table to be visible to other desktops, that is, thread 'a' needs to obtain a handle to desktop 'B' with read access.

It appears logical, but needs to be clearly documented!