Flight Instructor on training flight with student reported a near miss with another aircraft during taxi out for takeoff.

Narrative

Acting as Pilot in Command during a PPL training flight; my student and I in a school C172 (Aircraft X) received taxi clearance to Runway XXL at ZZZ from the south ramp on the east side of the airport; adjacent to FBO. We were given clearance to taxi to Runway XXL via taxiway Charlie. Approximately one minute into our taxi along Taxiway (TWY) 1; a pilot of a Citabria (Aircraft Y) requested and received clearance to taxi to Runway XXL via the same TWY and was beginning taxi from the ramp adjacent to FBO. This ramp is north of the ramp Aircraft X began its taxi. I heard the clearance and became aware of an Aircraft possibly leaving the ramp along our route. ZZZ Ground Control did not issue any directions to give way for either aircraft. At one of the ramp exits for Aircraft Y; a near miss occurred when Aircraft Y began to enter TWY 1 without verifying it was clear of aircraft. My student immediately offset away from the other aircraft and applied brakes of Aircraft X. The pilot of Aircraft Y eventually did the same thing. The wingtips of each aircraft were approximately 20 feet apart. The pilot of Aircraft Y hailed ZZZ Ground about the clearance; asking that they thought they were clear and almost ran into a 172. The ZZZ Ground Controller responded that our aircraft was also northbound on TWY 1 and apologized. The flight continued without additional incident. Something I learned that can prevent this situation is proactively querying the controller if giving way would be necessary for either aircraft.