Air Carrier First Officer reported aircraft PACK's malfunction during cruise. In spite of communication and O2 mask issues the flight crew was able to descend and reestablish the aircraft to a stable condition and safely continue to destination.

Narrative

Just as we were reaching cruise at FL430; we got the first indication that we had a malfunction (overheat) with one of 2 PACK pressurization systems on the plane. Following the checklist allowed us to continue the flight at 430. Some 4-7 minutes later we developed a malfunction with our second PACK pressurization system; and noticed the Cabin Altitude began to climb. We donned O2 Masks and began an immediate descent. At the beginning of the descent we were getting a lot of static/interference on our intercom system; and I was mostly unable to hear ATC or my copilot; but we communicated with gestures and heading/altitude settings. Communication was further hindered by a malfunction of my O2 mask passing through about FL350. ATC gave us headings and altitudes to follow; and we complied as best as we could. The controller did an outstanding job. We had thought we had officially [requested priority handling]. After leveling off at 10;000 feet and restored normal communications in the cockpit we were surprised when a new ATC controller asked if we would like to [request priority handling]. At that point the priority portion of our flight was behind us; the aircraft was in a safe and stable condition; and passengers were unharmed. We declined [requesting priority handling] at that moment. In hindsight; perhaps we should have anyway.Cascading mechanical issues. Primary issue caused within the pressurization system. Complicated by an improperly connected O2 mask; which hindered focus and communication. Primarily caused by a maintenance event; not sure that's avoidable. There might be a way to test the O2 masks so mine/others don't malfunction again.