Testing for Heterogeneous Impacts?

I've seen it many times before. After showing your baseline results, the thing to do in applied micro research is to split your sample--based on education, gender, age, etc.--to see if impacts vary across different populations. But let's say the estimated coefficients are about the same in the two samples, but one estimate is statistically different from zero while the other isn't. What to do then? Here's what not to do: Claim they are different, that there's an impact in one population but none in the other!!! Ok, so that's a typical mistake but nothing new. 

What I didn't know (but that seems so obvious after reading this twitter thread)? Finding one statistically significant impact in one population but not in another, even when effects are constant, is especially likely when effects are moderate, not too big and not too small. Check out the cool simulation of this here.