I'm gonna say not at all.
First, you have to modulate the source in such a way as to encode a message. I think we can rule out things that involve blocking the beam, so you'll have to adjust the generation power. You're gonna need a massively powerful nuclear reactor or particle accelerator or something to be at all possible to notice the message, so it will probably be pretty tough to modulate that much power at a frequency high enough to get any kind of decent data rate.
Then we need a detector. Since the article is about a massive and massively expensive detector being able to create sort of an image of the Sun after multiple years of observation, I'm not optimistic about that side. We can build a detector that can tell if a manmade beam is on or off, eventually. I'm not very optimistic about building a detector sensitive enough to detect subtle variations in the power of the beam. We're gonna have a real tough time getting a decent data rate.
Doesn't make much difference if 1 bit can be transmitted through the earth faster than an electric signal can make it around if the electronic one can send billions of bits in the time the neutrino detector takes to send two.