I have a video on Facebook that I'd like to share with people outside Facebook, but I don't have the original video file.
How can I download the Facebook video to put it on public web sites, like YouTube, so that others can view the video?
As far as Facebook, the web site, goes -- you can't. They don't provide any way to download the videos via the Facebook website that I know of.
One alternative is to use browser extensions that allow you to download videos from, in fact, any web page:
Download Helper (Firefox)
FVD Video Downloader (Chrome)
Note that for political reasons the FVD downloader won't work on YouTube, because the official Google Chrome add-in repository policies prevent it.
If you don't use Firefox or Chrome
First, try the Facebook Video Downloader User Script. User scripts work in most common browsers now. Click the big green "Install" button on the upper right of the userscripts.org page to set it up.
Another alternative, if you are using Windows, is software that pulls the video files out of the browser cache like VideoCacheView.
You can do the same thing manually if you navigate to where the browser cache files are on your OS / Browser, and sort by filesize.
using the Google Chrome browser
Here are the tutorials on how to download a video on Facebook:
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