I can only speak for myself... but my inner monologue is just a runaway train of thoughts that I vocalize in my head as I work through them. There is no other person in there, abstract or otherwise. Have you ever seen the Death Note anime? Similar to that.
I do what the person you're replying to and for me its just one voice or a faint imagination of them speaking. Its me roleplaying as the other person, not like there is a different entity in my brain.