This paper writes (edited for concision):
Consider, a doctor in Germany confronted by a meta-analysis of long term‚ $\beta$ blockade after myocardial infarction. Although a robust beneficial effect is seen in the overall analysis, in the only trial that recruited a substantial proportion of German patients, there was, if anything, a detrimental effect associated with $\beta$ blockers. Should the doctor give‚ blockers to German patients who have had an infarction? Common sense suggests that being German does not prevent a patient from obtaining benefit from $\beta$ blockade. Thus the best estimate of the outcome for German patients may come through discounting the trial carried out in German patients. This may seem paradoxical; indeed the statistical expression of this phenomenon is known as Stein's paradox.
Applying the findings from meta-analyses often means that the results from a particular trial are disregarded in favour of the combined result. This will generally be based on the assumption that inconsistent results are purely due to chance. But even if some real differences exist the overall estimate may still provide the best estimate of the effect in that group (Stein's paradox)
To me, this just seems like an example of applying shrinkage. In trying to frame this as Stein's paradox, I'm looking for some estimator for several parameters together that outperforms estimating those parameters individually. In this case the parameters might be (for sake of example) the subgroup effects for people in France, Germany, and Holland. So if I'm interested in a combined estimate in all three countries, I should use a Stein estimator.
However, if I'm interested only in Germany, I should just estimate the effect individually. The "best estimate" is only the best insofar as it minimizes mean-squared error for estimating all three countries. But given my exclusive interest in Germany, I might actually make my Germany estimate worse at the expense of improving estimates for countries I don't care about. I can see other reasons to apply shrinkage here (eg, there's very little data on Germany) but it doesn't seem to be addressing the crux of Stein's paradox.
Addendum (can be ignored)
Maybe I'm misunderstanding Stein's paradox more generally, and perhaps the example motivating my interest in the topic can highlight that misunderstanding. Suppose we're interested in identifying the most effective of 10 drugs to treat a disease. We estimate the effect size of each drug and recommend the drug with the largest. Great!
Next, for whatever reason, we're asked to actually recommend three of the 10 drugs. The naive statistician might continue reading down the list and recommend the second and third largest of our calculated effect sizes. But someone aware of Stein's paradox would change the estimation procedure to account for it. (I'm not sure on the details here...maybe producing Stein estimates for every combination of three drugs and choosing the combination with the highest average effect?)