Thank you for the detailed explanation!

Both "Be required to" and "need to" express necessity. The difference is that "be required to" is more passive than "need to" and it is a bit more formal in use. "Be required to" suggests that there is an external pressure on the subject of the sentence that is not mentioned in the sentence. Using your examples: 1) He was required to carry the lady's bags - we are not told why, or who wanted/made him carry the bag, all we know is that he had to carry the lady's bag. 2) They were required to turn of their phones - they may not have wanted to turn off their phones but they had to turn off their phones because someone else or something else needed/wanted them to turn off their phones. "Needed to" is a bit more flexible because it can be used in situations of internal or external pressures, but tilts more towards internal pressures. Example: 1) They needed to turn of their phones. Maybe they felt is was necessary because they didn't want to be disturbed from studying. 2) I suddenly needed to vacate my desk. Maybe someone suddenly spilled coffee on your desk and you felt the need to vacate the desk. I hope this helps. ^ ^