If you want to record a log of all messages sent then use Journal Queues. Queue journals record a copy of the messages that have been removed from the queue. You then need to write a process to read the journal messages and store the messages, be that in a database or LogStash.

The advantage of using Journal Queues is that logging can be done asychronously. The disadvantage is that reading messages from queue with journaling enabled is a bit slower, up to 20% slower.