I'm thinking about shelling out some cash for a private soon.
If I'm going to be doing so, I'm going to want to get my money's worth. What can I do to make sure I get the most out of it and what should I expect? Am I going to roll with the instructor and then he's going to give me pointers on what I need to fix or should I already come with specific questions? And how much of the session should be rolling with the instructor and how much should be strictly just doing the moves?
What I think is beat for a private is the instructor teaches you the moves and then puts you in positional drilling where you'll be able to learn when and how to pull off the moves 
I would come with a few specific positions or moves that give you problems in mind, but the instructor may already have a plan in mind. My professor asks "what do you have trouble with", rolls briefly starting with me in the position that I am concerned about, and then determines what best fits my abilities and then works with me on those moves/concepts. What has amazed me with privates is the professors ability to find a concept that I am missing that applies not only to the particular issue I was concerned about, but helps me with other situations/positions as well. I have found private lessons to be a very good investment
If its your own instructor then try and have a quick word with him a week before or something and ask him what's your biggest problem / what you should work on
Then you can think about it for a week ie specific things you notice in the position(s) and it prompts him to think about it too (instead of him just shooting from the hip on the day) 
Say the private is 1 hour. I think he can reasonably assess your game, habits, strengths and weaknesses after about 20 minutes of flow rolling. He should easily be able to make some adjustments and offer strategic and technical pointers that would greatly benefit your current level and style of movement. Additionally, you should be able to ask him what you want to learn - and he should be able to oblige in that area as well- with some techniques and drilling. 20/20/20 breakdown. One hour is actually a good amount of time to invest and be able to get a return out of for sure.
who is the private with?
you dont have to name names, but is it a famous guy? A guy known for a certain style? someone you speak to regularly, but are just going to take a private with?
My answer is dependent on the answers to questions like that.
The guy is my regular instructor and he is a 3rd degree black belt.
RoidsGracie - The guy is my regular instructor and he is a 3rd degree black belt.
does he roll with you in class? if he does, he knows your game, let him fix it and ask about something he does really well that you want to implement
gusto -RoidsGracie - The guy is my regular instructor and he is a 3rd degree black belt.
does he roll with you in class? if he does, he knows your game, let him fix it and ask about something he does really well that you want to implement
^ That being the situation, very sound advice^