Two aircraft were flying in opposite directions on the track between Mackay and Clermont. The inbound aircraft to Mackay was being held at 5,000 ft until the outbound aircraft had passed. The outbound aircraft was maintaining 4,000 ft. The controller estimated that the two aircraft would pass at about 21 NM from Mackay. The controller was using Distance Measuring Equipment (DME) reports from both aircraft to obtain the required separation before allowing further climb or descent. When the inbound aircraft reported at 20 DME, and the outbound aircraft reported at 25 DME, he allowed the inbound aircraft to descend. Just after passing 4,500 ft the pilot of the descending aircraft sighted the outbound aircraft straight ahead and slightly below. He took avoiding action and reported the event to ATC. It was subsequently determined that the pilot of the outbound aircraft had not been trained in the use of DME, and that the equipment in the aircraft was unreliable. When asked for DME distances he had provided dead reckoning distances obtained by map reading, which were incorrect.