I checked that at private doctor's place during my last pregnancy. I don't know the difference between 政府醫院 and 私家醫院. From my experience, my doctor put his hand/fist into my vaginal to check. I didn't like the feeling at all.
Actually 唔係痛,只是不太舒服. 應該不是整隻手放入去. I didn't see the process, I just felt that the doctor used his fist size to do the measurement.
Should be all right, 不用太驚.