Not unless individual operators did so.
When we ran the PBBS at the National Guard building at Fort Richardson we would often print out relevant messages during disaster situations. Sometimes these messages would contribute to the daily sitrep.
More often we would retransmit them to specific users so they could be alerted.
We had a couple of funny situations where the relevant users were on the internal email system but not on the PBBS, so sometimes we would retype the messages for them.
But the idea of logging, or "monitoring" like in broadcast radio is a fairly recent development, and is used mostly to assure advertisers that the spots they pay for are actually aired.