Here is what I'm used to do:
"Some objects have the same names, but are in different namespaces"
That's why namespaces exist.
"Would it be appropriate to create a second .h file to expose only the public API? "
You always should expose only the public API. But what means to expose public API? If it would be only to headers then, since public API relies on private API, the private API would be included by public API anyway. To expose a public API mark public functions/classes with a macro (which in case of Windows exports public functions to the symbol table; and probably it will be soon adopted by Unix systems). So you should define a macro like MYLIB_API or MYLIB_DECLSPEC, just check some existing libraries and MS declspec documentation. It is sufficient, usually non-public API will be kept in subdirectories so it doesn't attend library's user.
"Should their be a .h file per namespace or per object or some other scope?"
I prefer Java-style, one public class per header. I found that libs written in this way are far more clean and readable than those which are mixing file and structure names. But there are cases when I brake this rule, especially when it comes to templates. In such cases I give #warning message to not include header directly and carefully explain in comments what is going on.