Well, we don't know. Clearly this is something that would have to be resolved by a theory of quantum gravity.
My guess is that it's something along the lines with how classical statistical mechanics work. In statistical mechanics, if you talk about the behavior of a gas, you know that there is a microscopic description of the physics that is exactly reversible, where you can take any one configuration and infer every other configuration that ever happens. For example, if I take a room, and start with all of the gas compressed into one cubic centimeter up in one corner of the room, then this gas will spread to fill the room, eventually reaching equilibrium, where it's spread evenly throughout the room. But if I could gain perfect knowledge about the positions and momenta of every atom in the gas, I could calculate that it all started up in one cubic centimeter up in the corner, and exactly when that happened.
I suspect that the relationship between the stuff that falls into a black hole and the radiation that comes out must be like that: it's so randomized that it is incomprehensible how we could calculate exactly what fell in. But if we had the correct theory of quantum gravity, and perfect knowledge of the Hawking radiation, then the physics tells us that we should, in principle, be able to determine everything that fell in as well.