HoMM3 HD is a mod (add-on) for the game Heroes of Might and Magic 3.

This raises another question for me. If players in HotA play their own game and only exchange information at the end of their turn, how is it that they don't find the same heroes in the tavern? It would seem logical that the hero pool is simply divided in half. If so, is the division algorithm known? Is it possible to determine which heroes I can hire and which I cannot?

Yes, the pool is divided by the number of players.
And it seems to work quite clearly here.

Heroes are divided randomly so that there are approximately an equal number of heroes of each class and approximately an equal number of heroes in total.
(if the number of available heroes of one class is odd, then they will be divided between two players in favor of one player, and then the next class with an odd number of heroes will be divided in favor of the other player. That is, the total number of heroes may differ by a maximum of 1 hero)
Heroes are shuffled in this (random) way every turn.

In short, it is impossible to predict which hero will be available and which will not during HotA.