The files are stored locally on disk. If you have a backup solution already in place that allows you to restore the computer to a previous state, or to restore deleted files, then it's probably pretty straightforward.
If you don't have a recent working backup, there are third-party tools that might allow you to recover the deleted files, but at the point at which the space on the physical disk is overwritten, that data is gone. Your odds of success aren't great on that front, and the tools required are generally not cheap, but you might luck out. I don't have a specific recommendation, but a search for "utility to recover deleted files on <operating_system>" should probably get you on the right track.
The LSOs will probably be in one of these locations:
Windows XP: C:\Documents and Settings\username\Application Data\Macromedia\Flash Player\#SharedObjects\<Random Hash>
Windows Vista and Higher: C:\Users\username\AppData\Roaming\Macromedia\Flash Player\#SharedObjects\<Random Hash>
Mac: /Users/username/Library/Preferences/Macromedia/Flash Player/#SharedObjects/<Random Hash>
Linux: /home/username/.macromedia/Flash_Player/#SharedObjects/<Random Hash>
Chrome is a different beast, and stores it's Flash-related files in an unpredictable location in a way that's transparent to Flash. Searching the filesystem for #SharedObjects should get point you to the right physical location.
Thanks! I don't know if this will bear any fruit but your information about the exact folder storage area is definitely useful, if I can find some way of recovering it. Thanks for the help!
Yeah, no problem.
For what it's worth, there are a number of good solutions for hybrid online/offline backups these days. They're affordable, and the online service ensures that if you get some kind of ransomware that encrypts your disks, that you can still recover the data and/or have the service fedex you a backup drive overnight, and the local backup(s) ensure that the data is backed up quickly.
The one I use has saved my bacon a few times where I've accidentally deleted things that I've spent hours and hours on.