for the materials itself, it is required that the pipes must passed all kinds of tests including hydrostatic. it is certified by the manufacturer by issuing mill certificate. for the assembled or completed pipelines, (for the contractor's advantage whether this test is a pay item or not) it is wisely to conduct hydrostatic test before backfilling the pipelines. it is therefore the cost of this testing must be incorporated to the unit cost of the pipeline or mentioned on "other items" or "contractor's adjustments".
but there are some contracts has a pay item for this test known as "testing and commissioning". read the specifications and preamble of your contract specially on pipelines.
Definitely read your contract documents. Sometimes you have specific requiremnts for who must observe hydrostatic testing. Sometimes the client will pay for the first test but any subsequent retesting is on the contractor. Sometimes all testing costs are on the contractor. Almost never are all testing costs on the client (at minimum retesting is on contractor usually), as this would not drive the contractor to do anything more than keep fiddling with the system until he could get a passing test result. When we write contract docuemnts for pipeline projects the hydrostatic testing is almost always on the contractor and must be performed in the presence of our inspectors to verify comformance.
With regard to the comment, "it is wisely to conduct hydrostatic test before backfilling the pipelines", I believe it is ultimately of most value to know that the pipeline "tests" in the exact condition it will service for the Owner i.e. fully installed and backfilled. How else would one detect problems or damage that might conceivably be caused in backfill operations, etc.? That of course does not necessarily preclude any other reasonable diagnostic testing (with good intentions as I suspect you mean); however, it should also be remembered that at least long lengths of open trench can arguably in the present day also represent significant (Public/traffic etc.) safety and security hazards, and additionally the pipeline can be temporarily much more vulnerable to flotation (as in flooding) and maybe even deleterious thermal or pressure-induced movement hazards if not provided with backfill stabilization. I would recommend perusal of minimum consensus standards for installation and testing of pipelines, e.g. such as ANSI/AWWA C600, Installation of Ductile-Iron Water Mains and Their Appurtenances (or whatever other standards are applicable for the specific pipe/service involved).