ASME Section IX says "....the welder shall be tested under the full supervision and control of the manufacturer, contractor, assembler, or installer during the production of these weldments."
My position would be that the welder assigned to supervise the test would have to have the authority to do so assigned by management.
There is no requirement within Section IX, or D1.1 for that matter, that requires the individual assigned with the responsibility for testing welders be qualified or certified as a welding inspector.
If the contractor or manufacturer holds a code stamp, they have to have a QA/QC manual that includes job descriptions and assigned responsibilities. One of those responsibilities is the qualification of welding procedures and welders. The QA manual is reviewed by the insurer and ASME. Once approved, the contractor or manufacturer is expected to abide by the manual. The AI, if there is one, is responsible to ensure the contractor or manufacturer implements the QA manual as approved. Any changes must be approved before the changes can be implemented.
To put this in perspective, I had an opportunity to test a pipe welder for a structural application a while back. At the same time the contractor was testing new hires for pipe welding. The shop janitor/foreman was in charge of testing the welders. After bending a set of specimens he noted a small tear in one of the bent specimens. He promptly took it to the belt sander, removed all traces of the tear and pronounced the specimen "Good enough!"
There are good contractors and there are a few I would rather not see on any of my projects. It pays to review the contractor's QA manual before awarding the contract. It is also advisable to accept simple fact that that you only get what you ask for and what you enforce. A qualified third party inspector can be helpful when it comes time to verify the contractor is abiding by the project specification as well as their own QA manual.
Best regards - A