All pilots and aircrew are prone to over-claiming. Air gunners are going to over-claim simply because multiple gunners are likely to be claiming to have destroyed the same aircraft, and pilots are going to over-claim because they see an aircraft badly damaged and they assume it's going to be lost; these are not "lies" in any meaningful sense. These factors are, incidentally, part of the reason eyewitness evidence is not considered particularly reliable (many of the inmates exonerated from death sentences had been put there by eyewitnesses who were certain and probably not actively lying). Add to the basic issues with human memory and perceptions the likelihood that plausible claims would get a positive reaction, either unofficially (from congratulations from colleagues to drinks at a bar to a nice evening with a person of the appropriate sex) or officially (say, a Knight's Cross) and there are even more reason to conclude that damaged aircraft must have crashed.
From a strictly practical position, I'd also think it would be unwise to instruct your pilots to chase an enemy plane to ensure that it was definitely destroyed if it's diving out of combat with some evident damage (like flames), as this could be both excessively risky for your pilots and prevent your pilot from combating a second threat. (this, not some sort of chivalry is why I'd tend to discourage shooting pilots* when they're parachuting to the ground: they are no longer an eminent threat, but the enemy force may have other aircraft that are current threats.)
Obviously, military planners need to know how many enemy aircraft were actually shot down, but they a) don't necessarily have a good value for this (it's easy to count wrecks on territory you control, but challenging to similarly verify claims resulting from actions enemy-controlled territory or, say, an ocean) or b) they may feel the need to overstate success for civilian morale (something far less fragile than most military planners assumed, especially in the democracies).
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* Discourage, not ban, at least over enemy territory. Over friendly territory, well, captured pilots could be useful sources of intelligence. I'll leave morality out of this, for now. Rather obviously, killing an enemy pilot when he's parachuting from a destroyed aircraft is distasteful, probably even immoral.