In the U.K. and Ireland:
The piling contractor installs and tests (or pays a testing specialist to test) the piles. He should also provide records of the work he is doing. The Geotech engineer employed to advise the Client's engineer will monitor the workmanship on site during construction, the production of records. He will also monitor the testing processes.
The question of certification of the records may vary in the U.S.A., and the particular records prepared will vary with the type of pile being installed.
Payment in accordance with the contract is usually re-measurable, and the contractor is paid for the total length of piles installed, unless he is on a lump sum. (In that case there will be a lot of special conditions, allocating the risk.) Measurement will normally be based on the lengths of piles recorded by the contractor in his installation records, and these will usually have been signed off by the Client's Engineer on the advice of his geotechnical advisor or by the geotechnical advisor himself.