No, radio signals do not go through the Earth. Not for any one reason, but for a series of reasons, many of which have been talked about here so I will make a comprehensive list, and there might be even more to add on!

1. **The Earth is quite dense:** As @Renan pointed out, x-rays can't even penetrate bone, and the photons in an x-ray are significantly more energetic than the ones in radio waves (plus the Earth is much denser than the human body). 

2. **Atmosphere:** The atmosphere in general will not keep signals inside the Earth, which is why we can communicate with satellites just fine. However, the waves coming off of a transmitter aren't really going in a straight line. This is only an approximation. In real life, most materials are [dispersive][1] so the light will move differently based on various aspects (check that link). If the conditions are just right, like the atmospheric region has the right water content, and the signal is sent at a certain frequency, one can get reflection from the atmosphere. This is called [over the horizon][2] (OTH) in the radar world. So now we have a signal travelling back down towards the Earth's surface. Will it penetrate? This is where it gets interesting.

3. **On the surface:** one of three things can happen, (a) the wave is heading towards water, (b) the wave is heading towards land or (c) the wave is heading towards an object (on land or water). The third case will either reflect the wave straight back up (then off the atmosphere and back to the receiver, etc) or will eventually become case (a) or (b) so we will focus on those.

* **(a):** if the wave is heading towards water, this is simple. At most microwave (subset of radio) frequencies water acts like a mirror, completely reflecting the wave. This is exactly why submarines are completely invisible to radar and one must use sonar to track them. **So no penetration**.

* **(b):** if the wave is going towards the Earth's surface and it is not water, it is likely to be soil. Most soils have a high concentration of metals (often iron, sometime magnesium). To the wave, these metals will resemble a perfect electrical conductor, just like the water. Although the ground may not be as smooth as the water so you would get some crazy scatter. But either way, **no penetration**.


  [1]: https://en.wikipedia.org/wiki/Dispersion_relation
  [2]: https://en.wikipedia.org/wiki/Over-the-horizon_radar