Helicopter Instructor reported taking evasive action to avoid a Regional Jet who also followed an RA resulted in a NMAC.

Narrative

The Helicopter was on a training mission with an instructor and commercial student on board; departing ZZZ in a left downwind departure Runway XX flow. The Helicopter started a climb up to 2800 feet in order to safely perform the vortex ring state maneuver. As the Helicopter exited the ZZZ Class D airspace the frequency was changed to CTAF of the local public airport ZZZ1; as the maneuver was supposed to be executed 2NM Northeast of the airport at altitude. As the Helicopter performed its first clearing turn no traffic was observed; in the second clearing turn a regional jet was observed north west of the Helicopter performing a steep climb and left hand turn at same altitude. The Helicopter turned southbound as evasive action and descended down to 2500 and the instructor decided to continue the maneuver ca. 2NM south east of ZZZ Airport in order to stay away from additional traffic on the same flight path.The instructor was informed after landing; that the regional jet followed a TCAS RA and took evasive action and the event has been categorized as a 'potentially significant' incident. Also the controller stated that they issued traffic advisories to us but we were already outside of their airspace and on a CTAF frequency; they also stated that no pilot deviation or violation is considered; the incident happened in class echo airspace.