Regarding the data from Spain, another way of estimating deaths from COVID-19 that does not rely on COVID-19 testing is to track the total number of deaths in a region (from any cause), and compare to the historical baseline. This measure of "excess deaths" can help see how many deaths are potentially being missed.
Of course, there are some limitations to excess deaths. There are some arguments that excess deaths could overstate COVID-19 deaths. For example, hospitals have seen
admissions for heart attacks and strokes sharply decrease during the pandemic, so some excess deaths could be due to people avoiding medical care and dying from preventable conditions at home. However, there are also reasons to think that excess deaths could understate COVID-19 deaths. For example, with fewer people on the roads, there have been
fewer fatal car accidents during the pandemic, which should decrease the baseline number of expected deaths.
With these caveats in mind, here are some data from
The Economist's site that is tracking excess deaths across various locales:
These data fairly clearly show excess deaths leveling off to baseline around May in Spain, so reports of a low number of COVID-19 deaths in Spain are plausible. Remember that Spain's lockdown measures were fairly strict (the Financial Times has a headline saying "
Spain's reopening is stricter than America's Coronavirus lockdown"), and other countries with very strict lockdowns have managed to reduce cases to near zero (e.g. China).
Regarding whether Spain is under-reporting cases:
The number of COVID-19 deaths reported vs the total excess deaths is not overtly different than other similar countries. However, the time period for the data in Spain (Mar 10-May 18) do not seem to include the time period for which the numbers were revised (around May 27-31 according to
@PeroK's post), so we may have to wait a few weeks to see whether Spain's revisions to their death counts were appropriate.