The crossing is not at a straightforward conception of "the end of time" in an expanding universe, since most possible observers are carried away from the final fall-in by the expansion of the universe, so there's nobody orbiting "at infinity" who could in principle see the infall take "an infinite time".
Horizons are part of the causal structure of the entire universe, black holes, planets, toads, warts, and all. The horizon is dominated by the central mass and spin, but not fully determined by it. The horizon in a close black hole binary (or triple) gets very complicated. ("The horizon" is not even necessarily physically measurable, and with black hole evaporation might not even exist, although there are other features which can be indicative of the point of no return for an infaller).
Preliminaries done, there is the "no drama" conjecture. Given a large enough black hole in a quiet enough setting a freely-falling infaller will not know she or he has passed the point of no return, perhaps for several minutes according to his or her wristwatch.
That's because the tidal curvature at the point of no return gets very small as we take the mass of a slowly-spinning black hole above millions of stellar masses, and that's the curvature that's relevant in spaghettification, the leading cause of death of astronauts entering isolated black holes.
Of course, most of the black holes we have found are far from isolated (otherwise we probably wouldn't see them with current equipment), so an infaller is likely to be blasted apart by hard X-rays and superhot gas instead of falling straight in.
The observables for something strongly accelerating into a black hole for a faraway orbiting observer can be quite different; unlike for speed there is no maximum acceleration in relativity. One would have to find a limit to acceleration in the behaviour of matter. An astronaut is not going to survive anything like the acceleration needed to make much difference to the distant orbiting observer though.
The distant observer in the not-really-our-universe Schwarzschild model and seeing the infinitely-prolonged final infall is at rest with respect to the central mass. Different observers, e.g. ones shooting themselves into the same black hole, or hovering just above a different black hole, can see qualitatively different things.
Generically, outside observers will see a dimming and shrinking of (practically) any infaller closer to the black hole than the observer. Many such observers will lose sight of the infaller before the infaller has truly hit a point of no return. Consequently some observers could find themselves seeing a presumed-lost astronaut grow brighter and bigger again, and leave the vincinity of the black hole. (Substitute gas, dust, and parts of stars for astronaut in the previous sentence, and that is what the Event Horizon Telescope collaboration, among others, searches for.)
