To whom is the statement of Imam al-Shafi'i and Imam Abu Hanifah, 'if a hadith is authentic then it is my school', addressed, and may laypeople apply it?

Summary

The statement is addressed to the mujtahids of the school, not to anyone who comes across an authentic hadith. One who wishes to act on it must be confident the imam had not received the hadith, and must have mastery of hadith narrator science, texts, hidden defects, and jurisprudence with its methods of deriving rulings. Imam al-Nawawi stated that it does not mean anyone who sees a sound hadith may act on its apparent meaning, and that this condition is hard to meet. Laypeople without the necessary qualification are not intended by the statement.

Written in our own words from the original fatwa — check the source before acting on it.