The main thing to say is that the numerous popular articles which say that Hawking radiation is particle/anti-particle pairs in which one of each pair gets swallowed by the black hole are misleading. Hawking radiation is almost entirely electromagnetic radiation---that is, photons. You can still use the language of particle/anti-particle, but it is a bit misleading. So this answers most of your question.
Some further points are worth adding.
Hawking radiation comes from the region near a black hole, it is not produced 'throughout space' as you suggest. In this respect the situation is not very different from a star, where the emitted radiation comes from the outer layers. In the case of the black hole it is the region fairly near to but outside the event horizon that radiates.
Finally, Hawking radiation is extremely dim for black holes of ordinary size. For a black hole small enough to emit brightly, the emission process itself uses up the energy of the black hole and it evaporates. So overall one does not expect much Hawking radiation in total from all the black holes of the universe, compared to other sources of light such as stars (except there remains some room for uncertainty about this since it is not yet known very well what the distribution of small black holes may be).