Is it true that when your phone bill comes and someone's been calling you from a private number, you can see the private number on the bill??

No. Or at least I don't think so.

I can only see the people I've called but not the ones that have called me.

No. The phone company (internal dept) can access that info, but you the customer CANNOT. You get your phone bill and there is no way you can see a restricted or unlisted number.
The only times phone numbers appear on a phone bill is when you make a long distance call.