D01 TRACON Controller reported a busy session with another sector controller that led to communication issues and a TCAS RA/NMAC.

Narrative

Aircraft X was a BE9L off of BJC heading out the east departure gate. Denver was in a land north and west configuration; which limits our ability to keep satellite traffic low and tunnel under the arrivals. I climbed Aircraft X initially to 12;000 ft. to tunnel under the first arrival gate I needed to cross but decided that it would be much less impactful to the east departure sector if I climbed the aircraft all the way to the top of our airspace. I advised the aircraft to climb to FL230 and to expedite through 15;000 ft. to top Denver arrival traffic. The aircraft later called and said he was doing 2;000 ft. per minute and looking out that seemed to be fine. The AR2 controller had Aircraft Y regional jet that was on a converging course with Aircraft X; but I believed that there was plenty of time to get the BE9L on top of the regional. Another Controller forced a highlighted datablock of the Aircraft Y onto my scope; I already had the tag pulled up; at the time I believed this was the AR2 controller doing so to show me a potential conflict. As the aircraft got closer; I noticed that Aircraft Y was still showing almost 400 kts. and that the confliction was going to happen sooner than I expected. I still believed that AR2 saw the conflict because of the forced datablock that was sent to me. I dialed AR2 to tell that Controller to push down Aircraft Y early; but nobody answered the line.I then left my scope and walked most of the way over to the AR2 sector; and verbally told the Controller to push Aircraft Y down; but she didn't acknowledge. I then returned to my scope and pointed out the traffic to Aircraft X. He advised he had the aircraft in sight and I instructed him to maintain visual separation.I then tried again to call AR2 on the line to advise that I had her aircraft in sight and was maintaining visual but got no answer on the line. At that point I walked over again and got her attention. At that point the conflict alert was activating.I then realized that her line had not been forwarded correctly; and that she had never seen the conflict beforehand. It was another departure sector that forced the datablock onto my scope.Closest proximity that I saw was about 1 mile and 1-200 ft. vertically. The BE9L reported traffic no factor; but I believe the Aircraft Y told the AR2 Controller that they had an RA and were going to call it a NMAC.We have constant radio problems; causing abnormal configurations. Other sectors are nearly always using other sectors frequencies. I believe that this also translates into people not forwarding their inter-phones properly as well; because at times people are confused as to who is working what airspace and what frequencies they should be using. There are currently at least 1 frequency in each area that has been out of service for months; and those sectors will borrow or share frequencies with other sectors.In addition to that; we have a 2-area split which contributes to one hand not knowing what the other hand is doing. Departure sectors have look and go through arrival airspace; and are forced via the SIDs to use it; but arrival sectors do not have this. The result is that the arrival area doesn't pay attention to the departure traffic and will rarely take any action to avoid a loss of separation with departure traffic. At the end of this incident; I believe the AR2 Controller actually told the Aircraft Y to stop their descent which was the exact wrong action to take; because she expected the BE9L to be staying under her traffic. While we do tunnel aircraft as needed; it's not exactly a common practice; especially in a Denver converging situation.We have had numerous errors involving departure aircraft during converging operations; when arrival area controllers have expected something different than what the departure area has to do. The reality is that one entire area doesn't understand what the other area does most of the time; and it's unsafe.