How can I see what a private number is?
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.