Let us examine two examples of analysis.

1) Knowledge is justified true belief.
2) Goodness is enjoyable mental states.

George Moore has famously argued that in such analyses we are analyzing our concepts. We are trying to understand what basic concepts cobble up to form a non-basic concept such as the concept of knowledge and of good. Now, if this is true, then by combining the concepts of justification, of truthfulness, and of belief, then we are outputting a non-basic-concept. So if (1) was true, then this combination should output the concept “knowledge”, and similarly for sentence (2). However, this means these two sentences should not be surprising. We should not have reflective people rejecting them, as in fact they should be trivial.¹

Moore put it by saying that questions such as the following ones are always “open questions”, things people can and do coherently doubt: «Is goodness enjoyable mental states?» — it seems even that we would learn something if we answered “yes”. However, if the analysis is correct, then the following sentence seems to be equivalent to that previous one, though it, in turn, does not seem to be an open question: «Is goodness goodness?». This is the paradox of analysis: it seems most, if not all attempts of analyses, end up as open questions like that! So they cannot be proper analyses.

However, I think this is only true if one takes philosophy to be about concepts, and not about things or facts in the world. If one holds knowledge to be a real, objective phenomenon, and the concept of knowledge (and the word “knowledge”) to denote it, and if one does the same for justification, truth, and belief, then the paradox of analysis vanishes.

This becomes clear when one rewrites (1) and (2) to make this clear:

(1*) The objective phenomenon which our concept of knowledge is about is the same as the objective phenomenon which the combination of our concepts of justification, of truth, and of belief is about.
(2*) The objective phenomenon which our concept of goodness is about is the same as the objective phenomenon which the our concept of enjoyable mental state is about.

Since it is to be expected that two of our concepts, unbeknownst to us, are about the same phenomenon in the world, there is no problem with open questions.²

[Note, 12/07/2017: The important point here is that no paradox of analysis surfaces if one is attempting to do an extensional analysis of two terms. Perhaps it doesn’t even surface when one does a natural-extensional analysis (sameness of reference in all nomologically possible worlds). And not even during essential analysis (sameness of reference in all possible worlds). The problem is only there when one is undergoing intensional analysis, that is, sameness of meaning.]
Perhaps some other analyses deal with human-made concepts which do not really denote any property (even if emergent) of a state-of-affairs. I suppose we call such properties observer-relative. If that is the case, then how can analyses be informative?

I just read something in the first chapter of The Palgrave Handbook of Philosophical Method, and in it there was a very plausible answer. When it comes to concepts about observer-relative properties (this is my idea), then analyses work because they unearth “tacit knowledge” we have about that concept (this is the book’s idea). We can only find out what we would really attribute that property, its sort-of-necessary and sort-of-sufficient conditions, when considering a variety of non-obvious cases. A herculean effort, then, is necessary to discover the correct analysis; it is not obvious, for it is tacit. So this also explains away the paradox of analysis.
¹ That is, unless the content of our concepts is not transparent (i.e. reflexively accessible).

² One interesting point that can be derived from this discussion is that it is one thing to be a state of mental satisfaction, and quite another to be a state with intrinsic value (i.e. a good state), just like being a lump of clay is quite distinct from being a statue. In these cases, one supervenes on the other, or is grounded on the other, or is explained by the other, but the two are not the same.