Usually, we just look at the TP&V. If it was published before 1923, or says it is a reprint of a pre-1923 edition, that's all we have to do.
In other cases, we may look up library publication data to prove, say, that a book published in the U.S. without a copyright notice was indeed published in the years when a copyright notice was required. Or we may simply see that a particular text was published by the U.S. Government.
The bottom line is the question: if someone comes to us claiming to hold the copyright on a text, do we have proof to show that they're wrong?
Whatever proof or search we have to do, we then file it, either on paper or electronically, so that the proof will be available in 20 or 50 years' time, or whenever the challenge is made.