Why do we not know whether or not neutrinos are their own antiparticles? Plan an experiment as follows: A neutrino source provides only neutrinos, and a detector is sensitive only to antineutrinos. If you get a signal, that proves that neutrinos are their own antiparticles. If something as simple would work, I reckon it would have been done already. So why does this not work (or does it?), and we have to rely on neutrinoless double beta decay searches?
 A: Good point, such experiments have been done and they see nothing.  But there's something more. 
The neutrino source provides only left-handed neutrinos, their spin pointing against their direction of travel, because the weak interaction does that.
The detector can be sensitive only to right-handed antineutrinos, their spins pointing in their direction of travel, because the weak interaction does that.
We see nothing - but that could be just because the neutrinos have the wrong handedness, and not because of a neutrino/antineutrino difference.
So it's not proven, and we need the double-beta-decay experiments.  
A: We know that neutrinos and antineutrinos exist, and it's possible to tell the difference between them. For example in a charged current detector electron neutrinos produce this reaction:
$$ \nu + n \to e^- + p $$
while electron antineutrinos produce this reaction:
$$ \bar{\nu} + p \to e^+ + n $$
And detectors can easily distinguish between electrons and positrons so they can easily tell neutrinos and antineutrinos apart.
In your experiment you emit a beam of neutrinos, and the detector would detect only neutrinos not antineutrinos, but that isn't proof the two are different particles. The neutrinos your detector emits all have left handed chirality while antineutrinos all have right handed chirality. You can change the chirality by having your detector move in the same direction as the emitted neutrinos at a speed faster than the neutrinos are travelling. That would mean the left handed neutrino in your frame would be right handed in the detector frame. If you did this and your detector started detecting antineutrinos you'd have proved the two are the same particle.
But for obvious reasons this is not a practical experiment. Because neutrinos are so light they travel at nearly the speed of light even when they have small kinetic energies. Designing an experiment where the detector moved faster than the neutrinos would be challenging at best!
A: Because the only way to be sure is to nuke it from orbit is to observe them annihilate each other.
The traditional way for this is to take some particles, preferably at rest, or at least in a beam, and some anti-particles, and put/throw them together. You then get to observe them annihilating each other, emitting two photons with the restmass of the particle/anti-particle.
However, this is very hard to do for neutrinos. You can not effectively stop them (at least not with less than a light-year or so of lead). You can barely focus them into a beam (only by focusing particles into a beam which then decay into neutrinos). In general you barely get them to interact with ordinary material, let alone with each other (there are trillions of neutrinos passing through your body every second, the vast majority of which never interact with any part of your body).
In other words, what you need is a process where you know that two neutrinos should be coming out (two neutrinos, not one neutrino and one anti-neutrino) and then observe less of those processes than you would naively assume if the neutrinos were not their own anti-particle. Because if they are, then they can annihilate before you can observe them. (Actually, the quantum mechanical amplitudes interfere destructively, so they don't even generate photons with the restmass, they become entirely virtual particles.)
One of these processes (the easiest to understand theoretically and the one that happens most often in nature, because of all those radioactive decays going on) is the double-beta decay, where two electrons, and two anti-electron-neutrinos, are emitted.
