Shadow has a similar level of anonymity as cryptonote coins without the bloat and scaling problems that they have.
That's second part is nonsense. It uses the same methods and has the same bloat and scaling problems (assuming they are actually problems that aren't solved by ongoing improvements to technology), albeit with reduction in signature size by a bit less than a factor of two.
Is it? Shadow blockchain is about 400mb. How does that compare to Monero and Aeon?
The difference is all about usage. There is no scaling/bloat advantage to Shadowcash other than the LLW signatures that are less than a fixed factor of two smaller on large mix factors.
EDIT: Oh and I forgot, Shadow uses 1235 denominations instead of 123456789 so there are more signatures needed for most payments. That should somewhat offset the LLW and leave the two almost identical in size.
So for the same size tx Shadow will have a larger mixing set size giving it better anonymity.
No. The LLW signatures reduce the tx size but the reduced denomination set increases it. Probably ends up the same or larger.
And how large is the Monero blockchain?
2-3 GB I think. The difference is accounted for by higher usage, not any difference in the technology.
Is that higher usage in Monero is mainly made up of small mining tx or dust?
There is a combination. I don't know what they are necessarily (and it wouldn't be a very anonymous coin if I did!).
Realistically Monero has been a top 20 coin for almost its entire lifetime and has had a good volume of trading on multiple exchanges (current daily volume $71000) along with some other activity, so it isn't surprising that it would have much more transaction activity than SDC (currently #55, daily volume $824). Seems obvious to me.
There was also a bug in the pool that was fixed in 2014 and spammed some crap on the chain, but that doesn't matter going forward.
Monero 2-3 GB vs Shadows 400MB. I would think that it is a difference in technology as to why Shadows blockchain is so much smaller, POS rather than POW, without even taking into account of the reduced ring sig size by a factor two.
As I already explained the ring size does not yield a significant size benefit because it is offset by the small set of denominations, so you need more signatures even if each is smaller. That does not explain the difference in chain size.