Instructor pilot reported an NMAC while departing a non-towered airport.

Narrative

I was the instructor onboard when we experienced a near miss during the initial climb phase of our flight. Before getting on the runway; I ensured that both the approach end and departure end of the runway were clear of traffic and noted that no other aircraft were announcing their intentions to overfly or cross the field. I also made sure my student did the standard CTAF radio calls for holding short; takeoff as well as our straight-out northbound departure on the departure leg. During our initial climb; my student made the departure call and shortly after just as we were leaving traffic pattern altitude my student started to make our northbound turn. Suddenly; I saw another aircraft heading straight at me and making a hard left turn. We got so close that I could clearly see their tail number. That aircraft then continued maneuvering; performing 360s near traffic pattern altitude at the departure end of the runway.Fortunately; both of our aircraft continued on without further conflict. In retrospect; this was a good learning lesson to also update my procedures. Although we followed correct radio communications and ensured the runway environment was clear; this incident highlighted how critical it is to ensure full situational awareness.One thing I could have done better was ensuring that my Sentry was actually turned on before takeoff. Had it been active; it might have provided earlier traffic awareness. This experience reinforced the importance of using every tool available particularly during departure when the workload is high.