Skylane 182/RG Turbo Skylane/RG
Flight Crew Instrument; Flight Crew Flight Instructor; Flight Crew Commercial
Flight Crew Last 90 Days 40; Flight Crew Total 402; Flight Crew Type 380
Communication Breakdown; Situational Awareness
ATC Issue All Types; Conflict NMAC; Deviation - Track / Heading All Types; Deviation / Discrepancy - Procedural Published Material / Policy; Deviation / Discrepancy - Procedural Clearance
Air Traffic Control Issued Advisory / Alert
Human Factors
We were 3rd in line for takeoff at ZZZ and it was particularly busy. There was a lot of rotary traffic present in the right pattern as I believe there was a helicopter event present. We had been having breakfast at the cafe and witnessed 4-5 Cessna 172's doing power off approach so they were in the left pattern. When we were holding short of [Runway] XX @ [Taxiway] A we could hear the Controller on tower was busy and we waited for around 10 minutes. In that time; myself and the PIC who was left seat agreed that it was standard traffic pattern; I checked the sectional to confirm. We made a comment that perhaps rotary traffic was using the right pattern for separation. We got called by Tower and the Controller thanked us for our patience and said Runway XX clear for takeoff. We proceeded to takeoff and climbed in the upwind to around 650 feet before making a turn to the left. Whilst we were in the crosswind segment; still climbing we noticed a plane that appeared to be 200-300 feet below us in right downwind for Runway XY off to our left; on its way to pass underneath us. It was probably 600 feet away when we first noticed it and it passed underneath our nose. We were both puzzled as to what they were doing. Once we were clear of them; we made another left turn into left downwind for XX the Controller came on the radio and said to us that we had come very close to 'my' ILS traffic for ZZZ1 and that they had it written down that we had requested right traffic; which was incorrect. Our inactions were verifying left or right traffic; assuming left traffic was standard especially due to all the rotary traffic using the right downwind for XX. The Controller was particularly busy and perhaps as we were heading to ZZZ1 they figured it was more direct to make right traffic. We climbed to 200 feet of TPA [Traffic Pattern Altitude] on the upwind so to make sure we were as close to TPA before being in Downwind at the wrong altitude.
We were cleared for takeoff for Runway XX.  Prior to takeoff; I was monitoring entry of all aircraft entering the traffic pattern.  Helicopters were on right downwind Runway XX; while fix-winged were entering left downwind Runway XX.  I never communicated which way I was going to turn crosswind since I was following the other aircraft that have been in the pattern throughout the morning.  After departure; I climbed to about 700 feet and started to monitor other traffic in the area prior to making my crosswind turn.  I saw a small aircraft to the left of me that was descending about 200-300 feet below.  I then started to make a left crosswind turn from XX0 degrees to approximately YY0 degrees.  I flew over the traffic and continued heading south for ZZZ2.  ZZZ Tower Controller then addressed me on the radio that they had me down for a right downwind turn.  I never once communicated I was making a right downwind turn nor heard the Controller mention; 'make right crosswind upon departure'.  If I wasn't supposed to make a left crosswind because of other traffic on an ILS for ZZZ1; why didn't the Controller mention that to me prior to departure?  I would have gladly complied and made a right crosswind turn and continued my course to ZZZ2.
GA pilots reported an NMAC occurred when they departed the traffic pattern using standard left traffic procedures; although the Tower Controller expected them to use right traffic.