EM doesn't have to spread spherically. There is such a thing as directional antennas, spotlights, or even lasers. Yes they spread too, but throwing a hard drive at distances on the scale of light years isn't going to be precise either. Go far enough and you won't hit the galaxy you wanted to.
Directional antennas still obey the inverse square law, because the beam still diverges as it travels. So the amount of energy that you spend sending the signal gets spread over a larger and larger area, meaning that at any given point in that beam, the signal is weaker and weaker, until it gets lost in the noise.
And even lasers diverge. An ideal laser has some minimal divergence that is inherent and cannot be overcome (if I understand correctly, it's because diffraction is non-zero even in vacuum). A true Bessel beam doesn't diverge - but it does not have a boundary, and hence requires infinite energy.
In order to achieve a Rayleigh length (effectively the "range") that is smaller than the distance from Earth to the nearest star, a (visible) laser should be approximately 40 kilometers wide. The effective range scales as the square of the beam waist, so you can communicate with the Andromeda galaxy using a laser just a little smaller than Neptune.
> Go far enough and you won't hit the galaxy you wanted to
This can be solved, in the limit of large distances (i.e. many galaxies away), by the aforementioned method of refueling/rebuilding/re-guiding along the way. But between each galaxy there's a vast, barren chasm to cross with little resources, I concur.
