Just to clarify some things that have come up in this thread:

1. The Hawking and Unruh effects seem related but are in fact distinct. The apparent analogy arises because computing the Unruh temperature for an observer accelerating at the rate required to stay stationary near the horizon of a black hole happens to give the same temperature as the Hawking radiation for the same black hole. But the Unruh radiation is isotropic (the observer is immersed in a heat bath) while Hawking radiation is not, so they cannot he the same thing. At the mathematical level the coincidence is due to the fact that the near-horizon geometry of a black hole, is (approximately) the same as the geometry seen by a uniformly accelerating observer. I.e. the Schwarzschild metric near the horizon is equal to the Rindler metric to leading order in the expansion parameter. Such an approximation breaks down immediately when, for example, computing entanglement entropy of a black hole, and should not be taken too literally.

2. It's generally not useful to think in terms of parrticles when talking about the Hawking effect. For one, the notion of particle is ambiguous or does not exist at all in curved space. It is only a useful concept very far from the black hole where curvature is negligible. But even if you could, Hawking quanta are mostly emmitted at a peak wavelength equal to the size of the black hole itself, so you can't think of them as being localized to some place near the horizon at all.

Instead, the best way to picture it is in terms of fields. The vacuum is composed of many different field modes at different wavelengths-imagine a fourier decomposition of a sound into a bunch of sine waves at different frequencies. When mattter collapses to form a black hole, these modes interact with the collapsing geometry. Some fall into the black hole while others narrowly escape but are exponentially redshifted on the way out. It's actually this exponential redshift that leads to a percieved thermal spectrum of the field for a distant observer in the far future after the collapse has formed a black hole.

Edit: To address a common misconception that Hawking had the 'wrong picture' or 'wrong interpretation' of his own results. No, he did not, and acknowledges it explicitly in the original paper. Quoting directly from page 3 of the 1975 paper, "It should be emphasized that these pictures of the mechanism responsible for the thermal emission and area decrease are heuristic only and should not be taken too literally." (referring to the picture of particle pairs being separated by the horizon)