correct.cpp

void Object::RegisterLua(lua_State *l){
	using namespace luabind;

	module(l, "Game")[
		class_<Object>("Object")
			.def("Show", (void(Object::*)(int))&Object::Show)
			.def("Show", (void(Object::*)(std::string))&Object::Show)
	];
}

object.cpp

void Object::Show(int v){
	std::cout << "Some integer: " << v << std::endl;
}
void Object::Show(std::string s){
	std::cout << "Some string: " << s << std::endl;
}
void Object::RegisterLua(lua_State *l){
	using namespace luabind;

	module(l, "Game")[
		class_<Object>("Object")
			.def("Show", (void(Object::*)(int)&Object::Show))
			.def("Show", (void(Object::*)(std::string)&Object::Show))
	];
}

object.h

void Show(int v);
void Show(std::string s);
///Register the object's lua module
static void RegisterLua(lua_State *l);