Private investigators are allowed to collect information such as phone records, headers mobile users?

NO.

Personal information is protected under the Privacy Act. In this case it is illegal to collect personal identifiable information to third parties in cases not covered by the law and by the competent bodies. 
The illegal collection of personal information constitutes serious violation of privacy criminally punishable. 
The individual concerned and Telephone Consumer holder has however the right to request personal information about your telephone traffic via request to the carrier.

For investigative advice please contact our detective agency Phersei in Milan.