Unlike ASME Section IX which requires a company employee (representative) to witness the qualifications, AWS D1.X allows the employer (contractor) to accept qualification by a previous employer, a third party, e.g., a testing lab, or by the current employer.
Interestingly, D1.1 does not use the term certification or certified in the text of clause 4. They simply say that the employer is responsible for the qualification of the welders, welding operators, and tack welders (reference 4.1.2.2).
The certifying statement at the bottom of the performance qualification record (also known as welder qualification test record, welder performance test record, etc.) is signed by the employer. I assume the employer accepts liability for the welder's work and signifies the tests were performed in accordance with the applicable welding standard with the certifying signature.
Best regards - Al