They take any faults [identified in themselves] and put them onto others.
It's fine, and it doesn't need "that were" to be grammatical or meaningful.
The bracketed element is a past-participial clause modifying "faults".
Participial clauses functioning as modifiers in noun phrase structure are semantically similar to relative clauses: compare any faults [that were identified in themselves].
Past-participial modifiers are bare passives, as evident from the admissibility of a by phrase: faults [identified in themselves by others].
But they are not classified as some kind of relative clause, since there is no possibility of them containing a relative phrase: faults which identified in themselves is grammatical but it has a different meaning.