Light Aircraft pilot reported traffic entering opposite direction nonstandard traffic pattern resulted in taking evasive action and an NMAC.

Narrative

The situation in question occurred as myself and a friend were departing ZZZ and flying westbound toward ZZZ1; as a flight of two aircraft. Preceding the situation; we had spent the morning flying in formation between nearby airports. Prior to landing at ZZZ; we noticed the CTAF radio frequency was very congested; with many pilots stepping on each others' transmissions. Flight visibility was unlimited; and the sky was largely clear with a scattered layer of clouds perhaps 4;000 AGL. The air mass was somewhat unstable; which contributed to moderate turbulence for our light aircraft; and consistent heaving thermal activity requiring near-constant power and pitch adjustments to maintain altitude. The winds on the surface were light and variable; so we chose to land on ZZZ's runway XX to reduce our taxi distance to the terminal building. Entering downwind; a glance at the windsock confirmed the variable wind was trending out of the north. We landed; taxied to the terminal. We discussed our plan to depart to the west toward ZZZ1; opting to utilize a left downwind departure from runway XY. As we entered the grass crosswind runway (XY/XZ) from the taxiway; I made a radio transmission stating 'ZZZ traffic; Aircraft X flight of two back-taxi runway XY; crossing runway XX. ZZZ.' I did not hear any transmissions relevant to ZZZ as we continued our back-taxi to position for takeoff. A moment later; I announced our intention to depart: 'ZZZ traffic; Aircraft X flight of two taking off runway XY; left downwind departure to the west. ZZZ.' We proceeded to depart runway XY; turned crosswind a bit below the prescribed 700 AGL; and entered a downwind leg parallel to runway XY as we continued our climb with the intention of leveling off at 2500 MSL. As we were entering downwind; another pilot; Aircraft Z; announced on CTAF that he was on a right base leg for runway XA in ZZZ. This was the first radio transmission we heard from Aircraft Z; and we were caught off guard by his decision to enter the pattern on a right base for a runway we perceived as having a tail wind component. I looked to my right and saw Aircraft Z; and announced that we had the traffic in sight and would continue our departure to the west. The inbound pilot did not acknowledge our presence. It appeared as though Aircraft Z was within 500 ft vertically of our altitude; and within 1/2 mile laterally as he lined up for final on runway XA. As the two-ship formation lead; the Aircraft Z passed behind me and below my altitude. Aircraft Y following me exhibits inferior climb performance to my Aircraft X; so I maneuvered in a gentle S-curve as I flew away from ZZZ to ensure both other aircraft had maintained separation. I saw Aircraft Z crossing over the threshold of runway XA; and while I did not make visual contact with Aircraft Y flying formation on me; I looked at the fields below and saw a second airplane shadow following my own. Satisfied that nobody had made contact; my friend and I continued flying westbound. In retrospect; the Aircraft Z pilot may not have heard either of my CTAF transmissions preceding our takeoff roll due to transmissions occurring simultaneously at other nearby airports; which may not have been audible to me on the ground as I keyed the push-to-talk button. Exacerbating the situation was his unconventional pattern entry; the shifting winds and atmospheric instability; and the lack of weather reporting at ZZZ.