Please look at the picture: you can see a neutrino detector being held. But what's also very important to detect neutrinos is shielding from every other type of radiation, so that you can differentiate between neutrino radiation and everything else. The shielding is done by metallic (likely, lead) bars around the detector. The shielding is much more massive than the detector.
Another point: in the article they say that they have detected a 136 neutrino events over 461 days. So you can transfer one bit ~10 days. Neutrino detector sensitivity scales linearly with volume. Now imagine if you scale down your ~5-10 liter detector to a volume of 0.2 liters (a fliphone volume).
http://www.sciencemag.org/news/2017/08/milk-jug-sized-detect...