A pilot and Check Airman on approach to an airport with PCT IFR flight following had a near miss with an aircraft apparently flying the same approach but not in communications with ATC.

Narrative

While flying the ILS 26 instrument approach during a checkride; our aircraft came very near another aircraft apparently flying the same instrument approach. Initially the Check Airman and I made visual contact with aircraft in front and above us. We reported the visual to Potomac Approach; which was providing flight following during our IFR flight. Potomac was not in contact with the other aircraft. The aircraft turned northbound; and I returned to my instrument scan; thinking that the Check Airman would keep visual contact. We saw on the Traffic Information Service display that the aircraft had turned back toward us; and was within 500 FT of our altitude; though we never regained visual contact. We eventually; apparently; passed almost directly beneath the other aircraft with less than 500 FT of separation. Although I queried Potomac for an update on the position of the aircraft; the Controller did not respond. I did not disengage the autopilot and start a descent until the Check Airman suggested I do so. I believe several factors contributed to this problem; including: 1. Lack of coordination in my aircraft regarding who would be responsible for maintaining visual contact with the other aircraft; 2. Lack of response by Potomac Approach when I queried for an update on the location of the other aircraft after losing visual contact; 3. The fact that the other aircraft was operating at an IFR altitude (although at 3;000 MSL they were less than 3;000 AGL) and conducting instrument approaches without being in contact with Potomac Approach; and 4. My mindset was that the Check Airman would help to keep me out of trouble. To prevent a recurrence; the main thing I can change is to treat the Check Airman as a non-pilot passenger; and not to expect him/her to keep me out of trouble. I believe I would have taken evasive action much sooner had I been flying solo; or had I not been in a situation where the person in the right seat was acting in the role of an evaluator or instructor.