Most of the dialysis centers that I have transported patients too, have been responsible for weighing their own patients. That is just simply how our contracts were written. If the contract required that the crew weigh the patient, I would certainly ensure that the crew understood that weighing the patient was part of their responsibility and also ensure that they knew how to do it. The other thing that I would expect from the facility end is that they'd instruct the crew in how to weigh the patient, using the facility's equipment, if the contract specified that the transport crew weighed the patient.
From a facility standpoint, I'd want to ensure that each patient is accurately weighed before and after therapy, using a standard procedure, and ensure that procedure is followed, regardless of who does the weighing. Ensuring that policy is followed would help minimize liability that could fall back on the facility.
No matter what, always be kind to the personnel at the various facilities you transport to or from. Because of crew attitudes, a company can gain or lose contracts. Since all transport companies can pretty much provide the same levels of service, it can very easily come to perceived crew attitudes. Lose enough transport contracts and you can kiss your employer bye-bye.