Where does Thunderbird, on OSX, store its offline folders? I marked my Sent folder to be used in offline mode, and asked the messages to be downloaded. I saw the status bar show my messages being downloaded. 

But I can't figure out where the messages were stored.

My profile folder doesn't have any new files, nor any that start with the word Sent:

~/Library/Thunderbird/Profiles/l5z5ge4s.default/Mail# ls -l Local\ Folder
# Nothing here named Sent.msf, or Sent.sbd


Where should I look for my downloaded messages?
I'd say you are looking at the right place.

I've configured Thunderbird to use IMAP to sync with Gmail. In that scenario, my sent emails are located at

/Users/&lt;username&gt;/Library/Thunderbird/Profiles/&lt;randomcharacters&gt;.default/ImapMail/imap.googlemail.com/\[Gmail\].sbd/Sent\ Mail


This SuperUser answer has a screenshot which shows how to look up the location for a particular account: