Thought experiment - would you notice if you fell into a black hole?
This is a great question, because it's a subtle variation on the usual question about spaghettification and supermassive black holes, and shows somewhat deeper thinking.
So let's assume the black hole is supermassive -- or more specifically that you are really tiny compared to the black hole -- so that we can ignore tidal effects. Tidal effects are the difference in gravitational "force" on two different parts of an object. In this case, I mean the difference between the acceleration of your feet and your head. Your feet are slightly closer to the center of the black hole, so they will experience a slightly greater acceleration than your head. You would feel this as a slight tug on your feet. The bigger the hole or the farther you are from it, the smaller the difference will be. At some point, it will be so small that it's "in the noise" and you don't even notice it. We're assuming that.
If your head were somehow stuck just outside the horizon,† you would be right. I don't think anyone would claim you wouldn't feel anything if your head were attached to a rocket keeping you out, while your feet dangled inside the black hole. :) But those aren't tidal effects; they're acceleration effects.
On the other hand, if you are falling into the supermassive black hole (even if you jumped off this crazy rocket just an instant earlier), things are very different. Your head and feet are being "accelerated" at basically the same rate (relative to some stationary coordinate system, let's say) because you are so small compared to the black hole. So your head is moving at roughly the same speed as your feet, which means that the signal doesn't have to actually move outward relative to these stationary coordinates (it can't). Instead, it just needs to move inward more slowly than your head. And that's entirely allowed everywhere, even well inside the black hole.
You'll typically see this sort of thing represented by a graph of the light cones. And inside the horizon, those light cones "tip over" towards the singularity. This means that even light pointed outward can't actually move outward; the outward-pointing light ray will still be moving toward the singularity. But your head (and your feet) are moving toward the singularity faster, so your head enters into the light cone of your feet. Which means that relative to your head light can still move outward, as can a nerve impulse. Basically, think of two light rays given off by your feet: one directed toward the singularity, and the other directed away from it. You'll probably believe that they have different speeds. The speed of your feet is somewhere between those two, as is the speed of your head.
So all that needs to happen is for your head to enter the future light cone of your feet before your head hits the singularity. Not a problem, since the black hole is so large and you've still got a while to go. Now, you might be concerned that your feet will hit the singularity before your head gets that first signal, which would seem weird. But then you remember that the concept of simultaneity is relative. Your head and feet are in the same reference frame -- at least far from the singularity -- so they experience things at basically the same rate, and nearly the same time as judged in their own reference frame.
† Just as a side note, you should try to distinguish between an event horizon and an apparent horizon. Technically, you're talking about the latter, which is the local surface where light rays that are directed outward can't actually move outward. An event (or absolute) horizon, on the other hand, has nothing to do with local effects -- at least not directly. You can only know if something is an event horizon if you know the entire future history of the universe. Unfortunately, the term "event horizon" is thrown around in popular descriptions of black holes when it shouldn't be. They happen to be the same for certain special black holes, but they really are different concepts, and the right way to think about a horizon is different in the two cases. I just use the term "horizon", and anyone who knows the difference will figure it out. A good (and accurate) popular reference for all such things is Thorne's "Black holes and time warps". The standard technical reference is Hawking & Ellis's "The large-scale structure of space-time".
A falling observer does not experience passing through an event horizon as you describe.
Instead a free-falling observer would see space-time as locally flat as long as the tidal forces were manageable. Your head and your feet are (nearly) sharing the same frame of reference.
The falling observer always sees the apparent horizon in front of them until they reach the singularity in a finite proper time.
Up to that point, as Mike correctly describes, because no stationary "shell" observers are allowed, and space itself moves towards the singularity, it is possible for both the following statements to be true.
Light directed outwards still travels at the speed of light outwards according to the falling observer.
That light will always end up at the singularity in the future, but after the observer gets there.
I find the following diagram helpful. It shows the world lines of head and feet in Eddington Finkelstein coordinates and I obtained it here. In this diagram the singularity and the event horizon are shown as vertical lines. The curved solid lines are the world lines of your head and feet respectively. Light cones are shown and these are limited by the trajectories of light directed radially inwards or outwards. Far from the black hole these would just be lines at $\pm 45^{\circ}$. At the event horizon the outgoing side of the light cone is vertical. Inside the event horizon we see that the future light cone is directed inwards and that nothing can escape to outside the event horizon.
Now follow what happens when the "feet" signal to the "head" by following the world line of an outging photon (the right hand side of the light cone). You can see that a signal from the feet is always able to reach the head right up until the head meets the singularity. But of course that outgoing signal never makes it out of the black hole, it too reaches the singularity sometime after the "head" does.
Suppose you are moving toward the event horizon at .99999999 c.
Your feet cross the horizon. No signal can leave your feet and reach your head, if your head stays outside the horizon. You are defeeted. Or are you?
In a fraction of a fraction of a second, your head has crossed the horizon as well. The photons that your feet sent are passed by the head, you notice nothing.
But wait! Why are we crossing the horizon so fast? Why not just mosey on over it, instead?
The horizon we are approaching is pulling space inward at a ridiculous pace. For a black hole to be flat at the event horizon, it has to be huge, and if it is huge, the rate at which it "pulls space in" gets ridiculous. (well, more like the radius over which its "pull space in at a ridiculous rate" is large: all black holes "pull space in" ridiculously fast if you are near enough to the event horizon).
In order to prevent ourselves from crossing the horizon at a ridiculous pace, we have to accelerate away from the black hole. But acceleration itself generates an apparent horizon. If we accelerate fast enough to "hover" crossing the edge of the black hole, we'll end up putting an apparent horizon between our head and our feet: you will be torn apart, but you'd be torn apart if you did it in empty space.
Space near the event horizon, if viewed to be stationary, has an event horizon sweeping through it at the speed of light. In order to keep ahead of it, you need to accelerate fast enough that events "near" the horizon never reach you -- generate an apparent horizon between you and it. Because if events "near" the horizon reach you, so does the horizon! If your feet are danging over this apparent horizon, no force will allow them to communicate with you. Go to empty space, accelerate in the same manner, and your feet still get disconnected from you causally. (the difference being, if you stop accelerating in empty space, you are now next to your torn-off feet. If you stop accelerating near the black hole, you cross the event horizon, and are now next to your torn-off feet.)
If you "stand still" near the event horizon, you don't feel anything as it sweeps over you. Signals from the part of your body that cross over first are sent, don't cross the event horizon, but they do reach the other side of your body -- after that part of your body crosses the event horizon.
The reduction in tidal forces (how hard it is to notice the black hole's gravity) and to the rate of acceleration required to keep ahead of the event horizon when near it are both functions of the size and mass of the black hole. And in the limit, an infinite mass black hole looks like the future: coming at you at the speed of light, uniform over all of space and time, and no way to go the other way. Your feet cannot send messages to your head in the present time, but they can send messages to your head in the future, both plummet futureward.