As @Laure mentioned, en means some of them and les means them (presumably all of them).
Ceci dit, vu la valeur de ces pierres, nous ne pouvons nous permettre de les distribuer à droite et à gauche, chaque fois que quelqu'un nous les demande.
Grammatically, this sentence is just as correct as the other one, but the usage of les doesn't make sense. The problem is that "chaque fois" implies that the stones have been asked for multiple times, and you wouldn't be able to give them away multiple times. This makes the les problematic, especially the second one. The first one could maybe be understood as a partial distribution, because "distribuer" can mean "to retail" a wholesale quantity into smaller lots; but "demander les pierres" means to "ask for [all] the stones".
The sentence has different meaning with en but doesn't contain such a contradiction.
It would be fine if the sentence with les used "prêter" instead of "distribuer", because then you could assume that the stones are taken back after each time.