The quick response without all the nuances of the code:

Contractors welding to an ASME boiler and pressure vessel code have to qualify their welding procedure specifications (WPS) which usually involves welding a sample and recording the data and test results on a procedure qualification record (PQR).

The welder that welds the test sample is qualified if all the requisite tests are acceptable. The remaining welders then test following the qualified WPS. They must work within the parameters listed by the WPS.

The qualification of the WPS and welders must be under the supervision of the employer. Unlike AWS, ASME doesn't permit the contractor to send them to a third party to take their qualification test. That doesn't mean the contractor can't hire a third party to document the process and perform the requisite tests, i.e., NDE and/or mechanical tests.

The contractor "certifies" the WPS, PQR, and welder by signing and dating the documented attesting to their results of the testing accuracy of the data recorded.

Generally, the contractor cannot hire a "certified" welder off the street and put him or her to work without qualifying and certifying the individual to their qualified welding procedures.

ASME operates under the assumption the contractor has the necessary staff and infrastructure to support their manufacturing operations. In short, there is an employee that understand welding documentation and technology to properly apply the technology, code provisions, and manage said operations to ensure they are in compliance with the appropriate code requirements.

A means of ensuring the contractor in performing as required, stamps are issued to the contractor limiting the scope of the work they can perform after demonstrating they have the personnel with the necessary competencies to comply with the code. To ensure the system works the contractor must have an insurance carrier that employs and Authorized Inspector (AI) that verifies the design drawings and calculations meet the code requirements. The contractor also has an approved quality assurance manual that outlines how the contractor is going to control the work that meets code. The AI verifies the QA manual is implemented and the contractor is doing what was agreed to. The AI will perform a review of the calculations, drawings, materials of construction, manufacturing, and testing required by the code, including a visual examination and witnessing the hydrotest of each completed code compliant vessel or piping. In short, the AI's job is to protect the insurer from mavericks that take short cuts and don't comply with the code or their approved quality manual. For the most part, the system works.

That's it in a nut shell, the Cliff Notes of ASME.

Best regards - Al