If I understand your question correctly, no it is not required, it is only needed for clarification, as the verb "escribir" conjugated to "escriben" could refer to ellos, ellas, or ustedes. Think of the phrase "Te amo" (I love you). One could say "Yo te amo", but the yo is unnecessary as the only possible person that can be referenced with the conjugation "amo" is "yo". But in this sentence "Ellos nos escriben un libro", without the "Ellos" it would be unclear as to whether They (those guys) write us a book, or They (those gals) write us a book, or You all write us a book.
From what I know of this subject (and it's not much!)...
Ella le escribe un libro a él = She writes a book to him (specifiying who exactly she is writing the book to).
Ella le escribe un libro = She writes him a book (more general).
I think you would get away with both phrases unless you're specifying "him"
I hope this clears it up a bit!