C210 flight instructor reported an NMAC in the traffic pattern while conducting a practice instrument approach.

Narrative

After shooting a straight in practice RNAV XX approach we did a touch and go; flew to ZZZZZ for practice holds and then performed the practice RNAV XY circle to land for Runway XX at ZZZ. During the first approach to [Runway] XX we could hear the radio calls of 2-3 aircraft in the pattern and see them. I did hear a plane ask how can you hear this radio and the reply by someone else was 5 by 5. As normal the landing light of our aircraft was on during the initial XX approach; and I confirmed it on the XY approach. The PIC (Pilot in Command) (under the hood) announced our position as we reached the shoreline and communicated that we were on the XY approach and circle to land. At the circle to land altitude (700 ft.) I saw an aircraft to the left side and 100-200 above going the opposite direction. We turned to the right and continued downwind at the circling altitude and executed a full stop landing. We discussed the near miss. As we taxied for take-off I heard an aircraft in the pattern ask for a radio check; then we took off to ZZZ1 and landed.During our debrief at ZZZ1 we confirmed that neither of us heard the radio call of the aircraft departing ZZZ. I was looking out for aircraft as we descended to the circling height but did not see the aircraft until we were 1-2 miles from the end of Runway XY. The other aircraft did have flashing landing lights and was a low-wing two-seater.A contributing factor was also the low and bright sun angle to right on final and the shadows and trees to the left putting the opposing aircraft in the ground clutter. No idea how we both missed a call from the departing aircraft. We could hear aircraft traffic and the Tower clearly as we departed ZZZ and flew to ZZZ1. ZZZ is often used for IFR practice and training. There are two RNAV approaches that are useful in maintaining proficiency. The RNAV XY circle to land is particularly realistic and also does not fly low over homes. The airport is also used for primary training and has a popular restaurant. So not every pilot would be as familiar with the calls for practice approaches with circle to land as an option.