This study addresses contact-rich object insertion tasks under unstructured environments using a robot with a soft wrist, enabling safe contact interactions. For the unstructured environments, we assume that there are uncertainties in object grasp and hole pose and that the soft wrist pose cannot be directly measured. Recent methods employ learning approaches and force/torque sensors for contact localization; however, they require data collection in the real world. This study proposes a sim-to-real approach using a privileged training strategy. This method has two steps. 1) The teacher policy is trained to complete the task with sensor inputs and ground truth privileged information such as the peg pose, and then 2) the student encoder is trained with data produced from teacher policy rollouts to estimate the privileged information from sensor history. We performed sim-to-real experiments under grasp and hole pose uncertainties. This resulted in 100%, 95%, and 80% success rates for circular peg insertion with 0°, +5°, and -5° peg misalignments, respectively, and start positions randomly shifted ± 10 mm from a default position. Also, we tested the proposed method with a square peg that was never seen during training. Additional simulation evaluations revealed that using the privileged strategy improved success rates compared to training with only simulated sensor data. Our results demonstrate the advantage of using sim-to-real privileged training for soft robots, which has the potential to alleviate human engineering efforts for robotic assembly.