When you say "private" are you referring to a non-routable IP? Are you also asking this because you would like a callback coming back to your listener that is located on this "private" IP? I can only assume that this is for testing purposes, is that correct?

In your situation, I think you may be in need of port-forwarding. You would essentially set your LHOST to your non-routable IP Address, and note your LHOST and LPORT values which you'll be catching the payload on (Assuming a reverse connection type scenario). You'll need to login to your Router / NAT device and locate the port forwarding section and forward the LHOST and LPORT that you'll be catching the reverse connection on.

Lets say my non-route able IP address is 192.168.0.100 and my WAN IP is 300.400.300.400 (Hypothetically here folks - as if this is even possible anyway). You've created a meterpreter backdoor which utilizes a reverse connection payload, you'll set up that meterpreter backdoor to connect back to your WAP IP (Assigned by your ISP), on the given port. You'll configure port- forwarding up and allow the given port to be forwarded for 192.168.0.100 which is listening for connections as you specified.

In a bind scenario, their machine would listen on a given port and you would be able to connect to it. Their NAT device may interfere here if they have one setup because to connect to a certain port that isn't forwarded from the outside - a nmap scan on the port that may be open, may cause the results to look filtered if this device is in play.

This is a small piece of reason why the preferences are reverse-based vs bind-based in these type of attack situations. Toy around with it and if you have questions feel free to ask. Above is my understanding of it, if I'm wrong someone feel free to correct me!

-Kris