Their knowledge of each other's family background occurs within minutes of separating from each other after meeting at the feast at the Capulet's house.
As the nurse is talking to Juliet, Juliet tries to be rather coy about learning Romeo's identity. She points to several different guys in the room and asks their identities first. Then she tries to learn of Romeo's identity. This is just like a teenage girl trying not to give away her crush. However, when she learns that it's Romeo she gives it away to the nurse by saying that her grave is like to be her wedding bed. This is like a girl today saying, "Like, I'll totally die if I can't have him!" Now the nurse knows their secret and will have to remain in the secret. These are important features of their learning of each other's identities that will play important roles later.
Romeo and Juliet each find out who the other is at the end of Act I, Scene 5. This is the scene where the Capulets hold a dance and Romeo and Juliet first see each other.
Both Romeo and Juliet learn of the other's identity from the nurse. The nurse comes and tells Juliet her mother is looking for her. Romeo asks the nurse who Juliet's mom is and the nurse tells him.
Then, after Romeo has gone, Juliet asks her who Romeo was. The nurse tells her. Both Romeo and Juliet are unhappy, but still in love with each other.