I've emailed the TSA Offices today to get a definitive answer about detection and the subsequent procedure, with an emphasis on an individual's right to privacy.
We'll see what they say.

Ok, I got a reply. Despite me asking quite specific questions regarding tapes, glues, screening, detectability and a passenger's rights to privacy, I got a fairly genetic reply.

Regarding hairpieces specifically all they said was:

"If a passenger’s hairpiece sets off the metal detector alarm, the hairpiece must be inspected visually and the passenger’s head area must be searched with a limited patdown. If the Transportation Security Officer (TSO) determines that the hairpiece needs to be removed, the TSO must offer the person the opportunity to remove the hairpiece in a private screening area. If the individual declines the private screening, he or she must remove the hairpiece in the public screening area. We advise passengers to use non-metal hairpins to secure their hairpieces. Any passenger who does not permit the search will not be permitted to board an aircraft."
We hope this information is helpful.

TSA Contact Center

So, they obviously don't want to divulge what their scanners can & can't detect, for security reasons.
But I think their point is, Its not the wearing of a hairpiece thats a problem, its if its considered to be concealing a prohibited item.
If a scanner can successfully detect a piece, it would also reveal that it wasn't concealing something and therefore is of no interest to them.
Whatever the issue, it would have to be dealt with in private if it needed to be removed.
But reading between the lines, it sounds like a 'pat-down' could be conducted in a public place.
But ONLY if there was something to raise suspicion other than the hairpiece itself.