With my first, my private doctor in HK would take my weight and do a scan at every visit, but not necessarily take my blood pressure or urine sample at every visit. Scans are charged separate from the consultation, so most private doctors will do a scan at every visit.
With my third, my public doctor in HK would take my weight, blood pressure, and urine sample as well as monitor the baby's heartbeat using a doppler at every visit. Scans are scheduled as and when appropriate or necessary. There are no charges for any antenatal consultations, tests or scans in the public system.
I wasn't pregnant when I was in the US, but I do find the Hong Kong public health system pretty extraordinary. Private hospitals all send patients with major complications to the public hospitals, which are the only ones really equipped to handle them. Strangely, 60% of the medical staff is employed in the private system to treat only about 10% of the population, but the best care is actually found in the public system, especially at the teaching hospitals. Wish we have such a system in the States!