regarding israel, there is no general legal obligation to report a crime. just recently the head of the ethics in medicine committee in israel published a directive that hospital staff are not allowed to report or help the authorities even if there is suspicion of involvement in a crime. this came about after several instances where doctors and nurses detained patients on false pretences at the polices bidding, mostly when work immigrants are involved.
however there are a few exceptions in which it is obligatory to report a crime. one exception is the case of preventing a crime - israel's penal code states (roughly) "a person who knows about a persons intent to commit a crime and did not take all the reasonable measures to prevent its execution or completion (the crime), is punishable by up to two years in prison". this clause is hardly ever used, mostly because its hard to prove knowledge of a crime and because the authorities don't seek to punish based on this clause, but they did convict margalit har-sheffi of failing to prevent the murder of the late PM Yitshak Rabin based on this clause.
yet another exception is the obligation to report abuse of a helpless person (mostly children). the obligation to report only apllies if its a recent crime and if the abused is in the custody of the abuser. anyone is obligated to report such a crime as soon as possible, but the law also states that relevant professionals (medical staff, educators, social workers, police officers, clergy, etc) and the relatives of the abused are doubly obligated to report (they are subject to a prison term whcih is double in length then that of a regular person).
one very relevant issue is that of domestic abuse. there is an obligation of professionals (those listed above) to give information on how to contact the police or a domestic abuse treatment center to a person that they suspect is the victim of domestic violence. i'm bringing this up because i initially thought the obligation to report abuse included domestic abuse but as it turns out it only refers to professionals and only obligates them to supply information to the victim, not to report a crime.
while my answer refers only to the state in israel, the themes that come up here are the same themes that legislators negotiate in other countries including the states and the eu. namely, preventing crimes, protecting the helpless (and whether or not abused wives are helpless or no) and whether the entire public or just professionals are obligated to report.