I write up the majority of our SARs. I don't print them, so I don't have a paper log. I fill them out on my PC using Adobe, and I save it in a file aptly named "SARs filed". It is on my personal drive.
I email them to the BSA Officer (CO), who reviews them and mails them off. I don't know if she keeps a file, but that is up to her, and kept confidential if she does.
No log needed, really.
I, personally, don't mind getting "written up" by our Auditors. it tells me that we have a good Audit team, and it is usually to bring some mundane detail back to the surface for me to work on. They're happy, I'm happy, my boss is happy. The CO is happy, because we are then better prepared for the REAL auditors, the OCC.