Aircrew reported while on approach they observed on their ADS-B two proceeding aircraft come within 100 feet of each other while not communicating with each other. One aircraft landed safely and the other aircraft continued to operate near the airport with no communications.

Narrative

On Day 0; at about XA:30 pm local time; I was flying in Aircraft X on the RNAV 17 approach into TVY. I was #2 on approach. Another aircraft; Aircraft Y; was on the ILS 17 approach into TVY; and was #1 on approach. At about XA:30 pm I was crossing WEGET waypoint at 7000 ft. and 5 miles behind the #1 aircraft on approach; who was crossing JONEK waypoint and descending to meet the crossing restriction of 6100 ft. at JONEK. A third aircraft; Aircraft Z; was at JONEK in a circling attitude; climbing from about 5800 ft. up to an unknown altitude. Given the poor visibility; I couldn't not see either aircraft; however by use of ADS-B and the G1000 installed on my aircraft I saw the targets of Aircraft Z and Aircraft Y on approach merge at the same altitude. Fortunately the aircraft did not collide but I estimate they came within 100 ft. of each other. After the incident; Aircraft Z continued climbing and circling through the approach course for TVY Runway 17 approaches. The Aircraft Y on the ILS approach continued to make the approach into TVY. I made calls over the radio on 123.0 and 122.7 over the next hour looking for Aircraft Z; I received no response. I also monitored SLC Approach on 120.9 and 135.5 and heard no radio calls involving Aircraft Z. I'm unsure if Aircraft Z was monitoring any frequencies while operating near TVY. It should be noted that JONEK waypoint is 5.6 miles off the end of Runway 17 for TVY; and given the reduced visibility being 5 miles; Aircraft Z's flight behavior could be considered dangerous and even reckless. Aircraft Z continued to fly in sporadic maneuvers; crossing back and forth over the approach course for TVY at various altitudes from 5500 ft. to 7000 ft; making no radio calls to inform traffic in what he/she were doing. Had Aircraft Z been monitoring the correct frequencies; the pilot would have known that 2 aircraft were on approach into TVY and would have known to clear the approach course.