This article describes how to call a private number using five different methods.
In response to a mandate from the FCC to allow private calls, telephone companies developed a service called Last Call Return. This service automatically dials the last number that called your phone, regardless of whether the call was private or not.
The service is free for many, but not all, carriers. To activate it, dial *69 (in the US) from a landline or mobile phone. This only works for the most recent call, so make sure you dial it before another call comes in.
The disadvantages of this approach depend on the phone provider. For example, some providers provide a computerized voice that tells you the number, along with the option to make a call. Other phone providers call back to the private number and do not give you the number.