How can I see what a private number is? I've had a few missed calls from private numbers now, and I'd like to see who it is.

Is there a way to do this with a Galaxy S4 running Jellybean 4.2? 

I installed the app 'CIA caller' but that simply listed the private number as '-2', and when it looked up that number, it unsurprisingly said 'number not found'.

Any ideas how to get at these private numbers?
I don't think there is a good way to do this - the issue isn't that the private number is being hidden by your phone, it's that the phone company isn't transmitting the caller id info to you at all. So there shouldn't be any way for you to get that data working strictly from your phone.

Unless you know people who regularly block caller ID data, but don't leave messages, I'd just assume it's spam callers and ignore them.