Any Unknown or Unlisted Aircraft Manufacturer
Situational Awareness; Distraction; Communication Breakdown
ATC Issue All Types; Conflict Airborne Conflict
Flight Crew Took Evasive Action
Procedure; Environment - Non Weather Related; Human Factors
After getting the ATIS; requested taxi clearance from hanger/ramp on taxiway Romeo 2 to Taxiway Alpha for a departure southeast.  Tower replied; 'Present position; cleared for takeoff' without acknowledging the request to taxi to Alpha.  I told the pilot flying; that the Tower Controller didn't hear the request and to once again request taxi clearance to Alpha for the departure.  He did so and Tower cleared us to Alpha for takeoff to the southeast.  This is unusual; as they normally clear us to Alpha only; then once ready; we are to let them know we are ready for takeoff.  They will then clear us for takeoff from Alpha.  We taxied on out to Alpha and I told other pilot to get confirmation from the Tower that we were indeed actually cleared for takeoff.  He did so and Controller confirmed that we were cleared for takeoff from Alpha.  During this time; I noted that they were using both Runway X and Runway YY.  We departed Alpha; parallel Runway X and flew straight out and started our turn southeast as we passed abeam the Tower as is customary.  I heard the Tower Controller talking to an aircraft using Runway YY earlier and as we started our southeast turn; saw an aircraft climbing out from Runway YY on what would have been a collision course for us both.  Other pilot saw a different aircraft he pointed out; but I pointed out the one that was the immediate threat which surprised him as he had not seen it.  I told him to hold his altitude and turn hard right to pass behind the aircraft climbing out to the north from Runway YY.  After we were obviously clear now; I contacted the Tower and asked about us being cleared for takeoff with another aircraft coming off of Runway YY at the same time?  We had not been given any notification; advisory nor restriction concerning the Runway YY departure at any time.  The Tower Controller replied that the aircraft that we avoided would be 'coming back around' (assuming that meant staying in the pattern) and to maintain visual separation.  I told the Controller that we were now behind and would stay below that aircraft until clear to the southeast.  We passed well below him; now back southeast-bound; as he was establishing a downwind for Runway YY.  As I remember; I repeated my query as to the conflict to the tower; but there was no further comment concerning any of this from them.  We continued southeast for our transition training flight without issue; including working with the tower quite a bit later in the flight around Delta Taxiway; Runway YY and finally around the X area.  There were no further issues or comments and the training flight continued smoothly.    The ZZZ Tower has been doing some controller training and it is my understanding that the  Controller is a trainee.  This was evident during the early days over the last couple of months since Controller arrived; i.e. getting tongue-tied; or occasionally being corrected by the experienced Controller in the cab; etc.  This is to be expected and basically a non-issue as with any new person learning their job; while on the job.  There have been some frustrating times in that my students and myself have made multiple radio calls to the Tower with various controllers and either had a long delay prior to an answer; (with little or no other traffic at the airport) no answer at all requiring a second call; or several times; a change of radio to find out if the problem was in our aircraft or them for some reason not hearing us; or worst case; not paying attention.  During this same flight for the report today; we had requested an approach to and use of Taxiway Delta to do some ground reference work.  They cleared us on a right base for landing on Delta and other pilot answered their transmission as appropriate.  About a minute later; the Tower called us back to see if we could hear them.  Other pilot replied once again to their directions but neither of us can figure out if there was a radio issue in the  aircraft; or why they did not hear the first reply?  We heard both their calls loud and clear; and they apparently heard our reply loud and clear only on the second attempt.  This report is being submitted; not due to a deviation on my (our) part; but due to the potential collision hazard that the Runway YY departure presented with our own departure at nearly the same time without notification or restriction by the ZZZ Tower.  My general impression over the last couple of months; to include since the COVID-19 problem began; that there has often seems to be a lack of attention coming from the Tower as to communication and complete situational awareness.  I am hoping that by bringing this incident up in this format can somehow improve the situation so that it is a safer environment for all of us.
Pilot reported departing a runway and then observing an aircraft departing from a crossing runway causing an airborne conflict.