In an MLA relation, parties generally tend to use diplomatic means to communicate, to send and receive documents or evidence. However, this often takes a long time since there are more institutions involved. Therefore, direct communication between central authorities is highly recommended.
Some states assist each other in urgent situations through simpler means of communication like emails.
One other alternative is using the INTERPOL or similar criminal police networks for swift communication, which brings us to the next question.
 In a normal example, a request from a court in X state goes to the central authority, central authority sends it to the Ministry of Foreign Affairs and the Ministry of Foreign Affairs sends it to its diplomatic mission in Y state. The diplomatic mission sends it to the Ministry of Foreign Affairs of Y state and the Ministry of Foreign Affairs sends it to the central authority, which in turn send it to the actual practitioners for completion. The whole process repeats in reverse once the request is completed.