The Friar, like all Catholic priests who receive Confession, is sworn to secrecy. Does this mean the Friar must keep all Romeo’s secrets?
When a Catholic Priest receives confession, they are obligated by Canon Law (the law of the Roman Catholic Church) to uphold the Seal of the Confessional- a confidentiality agreement between the confessor and priest. Anything that is told during confession must be kept secret, even if someone has done or is planning to do something terrible.
This brings up a few thoughts- first, what about state law? If someone confesses to a crime in confession- say, a murder- the priest has a duty to hold this person accountable in the religious sense only. Intentional murder is a sin which totally alienates someone from God and the person cannot be redeemed. For the priest, they can only really council the person on how to stay out of further sin until their death. The priest might also encourage the person to turn themselves in to state authorities. The priest is not obligated by the Church to tell any state officials about the murder, but if questioned about his knowledge, he faces a difficult decision. In actuality, priests can and do break the Seal of the Confessional, but with penalty. A priest who breaks the Seal is excommunicated- essentially kicked out of the Church! In most parts of the world, there are laws offering priests protection on the understanding that they cannot break the Seal, but it is not unthinkable that a priest might be sent to jail for refusing to share information related to a crime.
As it applies to Romeo and Friar Laurence, anything told in confession must be kept secret, at risk of excommunication. Here's the thing, though- Romeo never goes to confession with Friar Laurence during the play! In Act II, Scene III, Romeo visits Friar Laurence in his living quarters and tells him all about falling in love with Juliet. Romeo beats around the bush a little bit and Friar Laurence does tell him to hurry up with his "confession," but they have not begun the ritualized format of the Sacrament of Reconciliation. In a religious confession, a person begins by saying, "Forgive me father, for I have sinned. My last confession was (however long) ago." They then go on to describe their sins and ask for forgiveness. No such exchange takes place between Romeo and the Friar during this play!
This ritualized format of the Sacrament of Reconciliation which is practiced today has its roots in the 9th Century CE, meaning that in the 16th Century, this practice would have been well under way. For Romeo's "confession" to take place in the way it does marks it as a non-religious conversation. In this case, Friar Laurence is under no religious obligation to keep Romeo's secrets. As a good friend to Romeo, he decides to, but as a religious official, there would be no risks for him to share the content of this conversation.