You don't have to orbit, you can just use a rocket to stay put. All observers that can communicate with infinity for all time agree about the infalling object. It gets frozen and redshifted at the horizon.
EDIT: in response to question
The issue of two objects falling in one after the other is adressed in this question: How does this thought experiment not rule out black holes? . The answers there are all wrong, except mine (this is not an arrogant statement, but a statement of an unfortunate fact).
When you are near a black hole, in order to stay in place, you need to accelerate away from the black hole. If you don't, you fall in. Whenever you accelerate, even in empty Minkowksi space, you see an acceleration event horizon behind you in the direction opposite your acceleration vector. This horizon is a big black wall that follows you around, and you can attribute the various effects you see in the accelerating frame, like the uniform gravitational field and the Unruh radiation, to this black-wall horizon that follows you around.
When you are very near a black hole, staying put, your acceleration horizon coincides with the event horizon, and there is no way to tell them apart locally. This is the equivalence principle, in the form that it takes in the region by the horizon where there is no significant curvature.
The near-horizon Rindler form of the metric allows you to translate any experiment you can do in the frame near a black hole to a flat space with an accelerating observer. So if you measure the local Hawking temperature, it coincides with the Unruh temperature. If you see an object fall and get redshifted, you would see the same thing in empty space, when accelerating.
The point is that the acceleration you need to avoid falling in is only determined globally, from the condition that you stay in communication with infinity. If you stop accelerating so that you see the particle cross the horizon, the moment you see the particle past the horizon, you've crossed yourself.
These two quotes seem like they contradict each other. Which one is correct?
They do contradict. Please be aware that comments cannot be downvoted so they often serve as a haven for content that an author suspects would be severely downvoted.
One thing that both the answer and the comment share is that the horizon is a lightlike surface. If a flash of light occurs below your feet then it reaches your feet before it reaches your head. It does not reach both at the same time. The event horizon, being also lightlike, follows that same pattern of motion locally.
This is true in every local inertial frame. The temporal ordering of lightlike separated events is frame invariant. So any nearby inertial observer, regardless of their relative velocity, will agree that the horizon reaches your feet first and then your head.
Best Answer
It depends on whether you are talking about everything you would see from that position, looking in that direction, or what the event horizon would look like.
In the former case, the answer is "No." Looking at a white hole, you would be looking inwards and all you would "see" would be the past singularity, which in the case of a static eternal black hole would be completely dark. Falling into a black hole and looking outwards you would see stars and galaxies. Completely different.
If you are talking about the appearance of event horizons before and after you get to them, then they all look the same, and they all look like ordinary empty space. Before you get to it, you cannot see it. It is in the future. To ask what the event horizon of a black hole looks like from the outside is like asking what 'next Tuesday' looks like. After you cross over you can see it (if there is anything there to see), and it looks like any other bit of past spacetime. The event horizon of a white hole looks like a patch of empty space between you and the past singularity. The event horizon of a black hole from the inside looks like a patch of empty space between you and the stars and galaxies.
Event horizons as such are nothing exotic. In fact, you are passing through countless event horizons all the time. The one-way nature of an event horizon is simply the one-way nature of the past moving through the present into the future. Nothing can escape from the future into the past, nothing that happens in the future can affect the past. It is exactly the same thing, locally. It's just that around a black hole, spacetime is twisted round so the entire future of points on the horizon is directed into the hole.
It helps a lot, in understanding black holes, to study the Rindler wedge.
This is the reference frame of a uniformly accelerated observer, like an astronaut on a continuously firing rocket. From the astronaut's point of view, the acceleration feels like gravity - it's very similar to the experience of an observer hovering a fixed distance outside a black hole (or sat on the surface of a planet). As the accelerating astronaut looks back at their own rocket exhaust, they see it receding ever faster, red-shifting into the darkness as the last few milliseconds of light are stretched out forever. The astronaut's coordinates have two event horizons - no matter how far in the future she goes, she can never see any event that happens after the future event horizon, and no matter how far in the past you go, an observer on the past horizon can never see the astronaut. The past horizon behaves like a white hole, the future horizon like a black hole.
And yet, for a non-accelerated observer watching the rocket zoom past, this is simply flat, empty Minkowski space! There is no gravity. There are no event horizons beyond which they can never see. This is much like the experience of an observer in freefall, dropping towards a black hole. To them, the event horizons are just undistinguished patches of empty space. And the reason no signal can escape is simply that signals cannot travel through any point from the future into the past.
White holes (in the static/eternal case) look black. Event horizons all look like ordinary empty space. Looking out past the horizon you see stars and galaxies. In fact, the appearance is almost exactly the same as what you would see just outside the horizon!