About the need of boson sampling verification
First of all, let me point out that it is not a strict necessity to verify the output of a boson sampler. By this, I don't mean to say that it is not useful or interesting to try and do so, but rather that it is in some sense more of a practical than a fundamental necessity.
I think you yourself put up a good argument for this when you write
Maybe a supercomputer powerful enough to calculate the permanent can do the trick. But then everyone would have to believe both the supercomputer's results and the Boson Sampling results.
Indeed, there are many instances in which one solves a problem and trusts a solution which cannot really be fully verified. I mean, forget quantum mechanics, just use your computer to multiply two huge numbers. You probably have a high confidence that the result you get is correct, but how do you verify it without using another computer?
More generally, trust in a device's results comes from a variety of things, such as knowledge of the inner working of the device, and unit testing of the device itself (that is, testing that it works correctly for the special instances that you can verify with some other method).
The problem of boson sampling certification is no different. We know that, at some point, we will not be able to fully verify the output of a boson sampler, but that does not mean that we will not be able to trust it. If the device is built with due thoroughness, and its output is verified for a variety of small instances, and other tests that one is able to carry out are all successful, then at some point one builds up enough trust in the device to make a quantum supremacy claim (or whatever else one wants to use the boson sampler for) meaningful.
Is there anything about BosonSampling that can be easily verified?
Yes, there are properties that can be verified. Due to the sampling nature of the problem, what people typically do is to rule out alternative models that might have generated the observed samples. For example, Aaronson and Arkhipov (1309.7460) showed that the BosonSampling distribution is far from the uniform distribution in total variation distance (with high probability over the Haar-random matrices inducing the distribution), and gave a protocol to efficiently distinguish the two distributions.
A more recent work showing how statistical signatures can be used to certify the boson sampling distribution against alternative hypotheses is (Walschaers et al. 2014).
All other works that I am aware of focus on certifying specific aspects of a boson sampler, rather than directly tackling the problem of finding alternative distributions which are far from the BosonSampling one for random interferometers.
More specifically, one can isolate two major possible sources of error in a boson sampling apparatus: those arising from incorrectly implementing the interferometer, and those arising from the input photons not being what they should (that is, totally indistinguishable).
The first case is (relatively) easy to handle because one can efficiently characterise an interferometer using single-photons.
However, certifying the indistinguishability of input photons is trickier. One idea to do this is to change the interferometer to a non-random one, such as the QFT interferometer, and see whether something can be efficiently verified in this simpler case. I won't try to add all the relevant references here, but this direction started with (Tichy et al. 2010, 2013).
Regarding the public verification aspect, there isn't anything done in this direction that I've heard of. I am also not sure whether it is even a particularly meaningful direction to explore: why should we require such a "high standard" of verification for a boson sampler, when for virtually any other kind of experiment we are satisfied with trusting the people doing the experiment to be good at what they are doing?