Does the radio (between two co-moving astronauts) stop working when crossing the event horizon?

If the astronauts are both infalling yet one (A) is closer to the singularity than the other (B), then their communication is already in trouble as they approach the horizon. Their different position in the gravitational field means that B is slowed relative to A. As the one further in crosses the horizon, his radio messages to his partner fade out -- smoothly, not suddenly -- because all their energy is lost. I don't think communication can be regained even after A crosses the horizon.

The radios would not work because the gravity gradient would rip the radios ( and the astronauts) into small pieces.

This is my second answer. Don't worry, it starts diverging from my first answer at some point.

Let's first consider two astronauts hovering near the event horizon of a black hole. When they communicate, using laser beams, they notice that they must aim the beams almost straight up, and that there is almost no delay, the beam is very fast. (In their accelerating frame the speed of light somewhere above them is billion cees)

An outside observer would say that it takes a very long time for the information to travel between the astronauts.

Next the astronauts turn off the rocket motors that made the hovering possible. Now things become normal for them. So it takes a normal time for the beam to traverse the distance.

Astronauts' very fast communication was outside observer's very slow communication, and now the communication slows down even more. The outside observer knows that the astronauts must be communicating extremely slowly now.

When the rocket motors were turned off, four things happened according to an outside observer:

1. Speed of communication jumped from very slow to extremely slow
2. The speed of the brains of the astronauts did not change
3. The speed of communication started accelerating
4. The speed of the brains of the astronauts started accelerating

All this happened because the astronauts were moving very fast when they were hovering next to the event horizon - which is often said to be moving at the speed of light. When they turned the motors off they started decelerating, and the ticking rate of their clocks and the communication speed started accelerating.

So, an outside observer says that radio waves can travel between the astronauts many times during the fall, because the radio waves travel quite fast between the astronauts, much faster than between astronauts that hover near event horizon.

Let's first consider two astronauts hovering near the event horizon of a black hole. When they communicate, using laser beams, they notice that they must aim the beams almost straight up, and that there is almost no delay, the beam is very fast. (In their accelerating frame the speed of light somewhere above them is billion cees)

An outside observer would say that it takes a very long time for the information to travel between the astronauts.

Next the astronauts turn off the rocket motors that made the hovering possible. Now things become normal for them. So it takes a normal time for the beam to traverse the distance.

Astronauts' very fast communication was outside observer's very slow communication, and now the communication slows down even more. The outside observer knows that the astronauts must be communicating extremely slowly now.

So it must be so that falling to the center of a black hole takes extremely long time according to an outside observer. That way the astronauts can have a quite long conversation while falling.

Thus, EM waves would not spread spherically anymore, but only towards the singularity

This is not true because inside the horizon "radially inward" is the only possible radial direction but the metric is still spherically symmetric, so the restriction is only abut the radial coordinate.

Does the radio (between two co-moving astronauts) stop working when crossing the event horizon?

Assuming that the black hole is massive enough that there are negligible tidal effects at the horizon then their radios would continue to work and their conversation would carry on without a pause.

Now there are others, who suggest that inside the horizon, everything, including light must move towards the singularity, the singularity becomes a point in time (future).

This is true also. There is no contradiction between the two claims. Because the astronauts are also falling in towards the singularity it is not necessary for light to go outward in order to go from one astronaut to the other. If you draw the worldlines of the communications you will find that indeed they never go outward.