Can a private caller be traced ?

Yes, but it will probably cost you.

I had 'private' or 'blocked number' or 'unknown number' calling my home at all hours. My phone company said that I needed to hit a certain code on the phone, *000 (cannot remember the numbers now), and the call would be flagged.

But, I did not have access to the records, but could request them through the police department.

In my case, I believe each call was $3.50.

In general no they can not be. If it is a legal issue, you can report them to the police