[I am hesitant to post this answer, as I am unclear that this is a question about science rather than philosophy as I said in a comment. But I've written it now, so.]
First of all, all scientific theories rest on a mass of assumptions not all of which are testable. For example if I'm doing astronomy I'm assuming that I'm not just living in a simulation, or that, in fact, anything outside my own mind exists at all. I can't test those ideas, but I'm going to assume them or I can't do any astronomy because there's no point.
Without assumptions like that all you can do is maths: Fermat's last theorem is true whether or not Fermat or Andrew Wiles, exist, say.
But there are assumptions which are made at a level above these kind of things which also matter in astronomy. For measuring the spatial curvature of the universe one is very important: the cosmological principle.
The cosmological principle. Viewed on a sufficiently large scale, the properties of the universe are the same for all observers.
(From William C. Keel (2007) The Road to Galaxy Formation (2nd ed.), Springer-Praxis, via Wikipedia.)
Here an 'observer' doesn't mean 'someone on Earth', it means 'someone anywhere you like'.
The cosmological principle implies two things: that on large enough scales the universe is homogeneous (it looks the same everywhere) and isotropic (it looks the same in all directions).
The cosmological principle is not fully testable I think. We can check that the universe looks isotropic here, and people do this. If it's not then the cosmological principle fails. But if it does look isotropic then the principle still may be false: perhaps we are in some special location where the universe looks isotropic. If we could get far away (very far away) and test that the universe looks isotropic from at least two other places as well, then we could conclude that the part of the universe we've observed is also homogeneous I think.
But it could still be fooling us somehow: the cosmological principle is really just something we assume is true. It's a very reasonable assumption, I think, because what it's really saying is that where we are is not special: the universe is not, in fact, built around us. The cosmological principle is the inverse of the Ptolemaic model, where everything goes around the Earth, really, and even more it is an inverse of the 'nothing but me exists' model which I said I was going to rule out at the start.
Well, if we assume that the cosmological principle is true, then the bit of the universe we can observe is just the same as all the other bits of it on large enough scales. So, if we assume that general relativity is a good description of the universe on large scales (that's a lot more testable) and probably some other things which are testable then if we can measure the spatial curvature (or equivalently, the matter density) of the part of the universe we can see, we can conclude that this holds for all of it.
That is the sense, I think, that claims about the testability of measurements of spatial curvature are true.
