1. A method of synchronizing contents of a first mailbox of a first messaging system with contents of a second mailbox of a second messaging system communicatively connected to the first messaging system, comprising the steps of: in response to receipt of each message for the second mailbox, the second messaging system storing the received message in the second mailbox and also making and sending a copy of the received message to the first messaging system for the first mailbox; in response to receipt of each message for the first mailbox, the first messaging system determining a source of the received message; in response to determining that the message was received from a source other than the second messaging system, storing the received message in the first mailbox and making and sending a copy of the received message to the second messaging system for the second mailbox; in response to determining that the message was received from the second messaging system and the second messaging system received the message from a source other than the first messaging system, storing the received message in the first mailbox without making and sending a copy thereof to the second mailbox; and in response to determining that the message was received from the second messaging system and the second messaging system received the message from the first messaging system, discarding the received message without storing it in the first mailbox and without making and sending a copy thereof to the second mailbox.