Well, let's say the original purchaser happens to be suspected of a crime (completely unrelated to a straw purchase, let's say bank robbery) and during a search of that purchaser's house, federal agents find a receipt for a check with "Glock 19" written on the memo line, it'll get their attention [0]. Very likely scenario :)

Other than that, good old fashioned surveillance plus a getting a confession [1]. I've also heard ads on the radio for a tipline to tell cops about straw purchases. No idea how well that works, though.

0 - https://en.wikipedia.org/wiki/Abramski_v._United_States#Fact...

1 - https://www.justice.gov/usao-edpa/pr/philadelphia-woman-conv...