Both Italy and Germany had pioneering jet programs, the German He178 and Italian Caproni Campini N.1 being the first jets to fly under only jet power. So did they ever exchange technology and research to further develop their jet research?
Yes, they did in fact share at least the theoretical bits of Jet technology. That said, it's interesting that Italy never actually made a jet powered fighter as opposed to Germany who actually went on to test a few other engines before arriving to the BMW 003 or the Junkers 004.
Probably the most important part is that the Caproni Campini N.1 was more of a hybrid plane that didn't utilize a turbojet design like the BMW or Jumo designs. Instead the Campini utilized a rather interesting hybrid engine that used a conventional piston engine to drive the supercharger.
What I will offer is that making jet powered fighters or planes outside of the realm of prototypes was the tricky if not outright difficult part. Like the Germans and Italians, the British, not too much later after the Italian prototype completed a jet engine in the form of the Whittle and a plane in the form of the Gloster E28. For the Italians, the difficulty of plane development remained a constant for aircraft as wartime resources made both precious metals and fuel rather scarce. Unlike Germany who had a much larger industrial base and (I would argue, until the later points of the war..) a much more avant garde aircraft design committee compared to both the Axis and Allies, Italy was constrained by its inability to manufacture large amounts of engines and airframes.
In short, I would say that the cooperation of Jet technology probably did exist, but considering how there are no records Italian forays into jet powered military fighters for mass production, I would wager that Italy had no resources to really develop or advance jet fighters.