If the universe were spatially of finite and not too large volume then we could, in principle, discover this by observations in the future. For example, there might be evidence of light setting off in opposite directions eventually arriving at the same point after travelling around the universe, or something like that. This can happen even if the average curvature is zero or negative if the topology has the required form.
However if the universe is very large then we will not be able to make observations like that. We will not be able to tell whether it is infinite or just very large.
In fact I think it fair to say that no experiment, even in principle, could establish beyond reasonable doubt that the universe is in fact spatially infinite. Certainly it is not something anyone can claim to know for sure.
What happens is that people working in cosmology find that many of the ideas do not require one to know whether or not the universe is infinite, so if one is studying the average properties one might just say "oh well let's just treat it as if it is infinite". It has become so common to do this that people often forget that this is not a hypothesis that has been tested at all. It is just a working assumption, or a way of avoiding the need for more information. But if you query this assumption then it is not at all clear whether or not it is right. The idea of infinity is reasonably well-defined in mathematics, but it is not clear whether physical stuff can be infinite. Do we really know? No we do not.
So the short answer to your question is "no-one knows". And the longer answer is "very likely no one will ever know."