PA-28 Cherokee/Archer/Dakota/Pillan/Warrior
Air Traffic Control Fully Certified
Communication Breakdown; Situational Awareness
ATC Issue All Types; Conflict Airborne Conflict
Air Traffic Control Issued Advisory / Alert
Human Factors; Procedure
Both aircraft were coming up from the coast and landing at the same airport; traveling along the same course; when the hand-offs were flashed to me; a BE35 was about 5 NM directly behind a PA-28 but with an appreciable overtake. I was immediately concerned whether he had the Cherokee in sight; ZZZ1 Sector XX was the transferring Controller and he/she initiated a call to me and advised that 'they have each other in sight.' I acknowledged; but was skeptical of his/her information since the one aircraft was directly behind the other; it's tough to keep someone in sight if he's on your six o'clock. The PA-28 checked in first and I asked if he was aware of the Bonanza on his 6 o'clock overtaking him at the same altitude. He replied no; the last he heard was that the other guy was 8 miles behind him. I gave him the position/distance/closure and he decided to maneuver right to avoid and let him pass; I acknowledged.  The BE35 checked in next and I asked him if he had the Cherokee in sight ahead at his same altitude; he replied no and; given the close proximity of my position call; he asked for vectors to avoid. Since the PA-28 was maneuvering right; I advised the BE35 to turn 20 left. After a moment on divergent courses; I gave other position reports on the traffic; but the BE35 still couldn't find the Cherokee even when he was 2 o'clock and a mile. The PA-28 finally got the Bonanza in sight on his left wing and they resumed their own navigation direct to the airport. Evidently ZZZ1 XX did issue a traffic call when the aircraft were 8 miles apart and got an acknowledgement; but acknowledging a traffic call is not the same as getting the traffic in sight; and even if the traffic was reported in sight (doubtful); it's no assurance that it remains in sight. Anytime two aircraft are closing to unsafe proximity; it's better to ensure separation than to assume they've got it covered; it was evident from the RADAR updates that neither aircraft was maneuvering off course to avoid the other. Even if the pilot previously reported the traffic in sight; once you see him closing on traffic and not moving then you need to take further action to ensure he's aware of what's developing. I suspect that the pilots took the 8 mile traffic call as a low-priority courtesy call (8 miles is a long way away) and they evidently were not aware of the overtake/closure aspect of the situation. They knew the other aircraft was out there somewhere; they just didn't know how close they were getting. Regardless of what ZZZ1XX had been told in his/her relief briefing; the Controller should have kept on the traffic calls when it became evident neither aircraft was avoiding the other by the time they got to me; it was rapidly becoming a safety alert. After I turned the second aircraft; I called the transferring Controller back to advise that the pilots did not have each other in sight. The Controller repeated what he/she was told; but it doesn't matter what you're told; it only matters what you do.
Enroute Controller described an unsafe event when visual separation advisories were not applied correctly by an adjacent controller.