The danger from gamma-ray bursts (GRBs) is the potential to reduce atmospheric ozone drastically, allowing harmful ultraviolet radiation to pass through the atmosphere and make life on Earth much more difficult. However, you have to be fairly close for this to happen. Gehrels et al. (2003) calculated that a typical GRB from a supernova would have to occur less than 8 parsecs from Earth to cause serious damage to the ozone layer - or, as they put it, to cause levels to drop low enough that they would
double the “biologically active” UV flux received at the surface
Obviously, though, we can detect gamma-rays from distances farther than 8 parsecs! The closest GRB detected so far, GRB 980425, is likely the result of SN 1998bw, and thus originated 140 million light-years away - over 42 million parsecs away. The radiation from it was nowhere near high enough to harm life on Earth.
Your second explanation, therefore, is the correct one. The intensity of the signal decreases according to the inverse-square law - that is, $I(r)\propto r^{-2}$ - so at large distances, the flux is small enough that there is negligible impact on Earth's atmosphere.
Phiteros also raised a good point: To cause severe problems from Earth, the gamma-ray emission from a GRB - which is often along an axis from both poles of the progenitor - would have to be pointed nearly directly at Earth. For example, there was once a scare that a supernova by either of two stars in the WR 104 system could cause problems for Earth, but the odds that the beam would hit Earth full-on are slim.