I coach a junior side, and as we have no officially appointed umpires, the coach or a parent from each team have the umpiring duties. Unfortunately this can create situations where some umpires are not impartial. I had a situation where my team was fielding, and the batsman got a thick edge to first slip, who fumbled the ball, but caught it on the second attempt, hugging it to his body close to the ground. The other teams umpire was at the bowlers end, and gave it not out, claiming he was unsighted so was unsure if the catch was taken cleanly. I was umpiring at square leg and saw the catch taken cleanly, so informed him of this, but he would not accept my word and refused to give it out. Can the square leg umpire give a batsmen out caught, or is the decision solely up to the bowlers end umpire? Similarly, can the square leg umpire give a batsman Not Out caught if he sees the ball is not taken cleanly, but the bowlers end umpire believes it was and gives him Out.
Umpires are taught to work as a team and one of the situations where this is crucial is the case you outline in your question. The bowling-end umpire, if unsighted, should ask for an opinion from the square-leg umpire.
The square-leg umpire cannot under any circumstances give a batter out caught. Nor can the square-leg umpire ever give a batter not out from an incompleted catch.
I appreciate your frustration but the bowling-end umpire was quite within his/her rights to give the batter not out.
The worst case I have ever seen was in a junior match many years ago when an appeal for a caught behind was turned down. The square-leg umpire (the coach of the fielding side) then walked towards the other umpire, told him the ball was clearly edged by the batter and ‘bullied’ the other umpire into giving the batter out.