I've installed the Fast Video Download version 3.0.8 addon for Firefox to download flash videos, like from youtube. What I'm wondering is, how does the addon download it, and do other people see that i'm downloading the videos? For example, is all the software to download the video already on my computer, or does the addon contact someone else to get the video, or let them know? Can the webpage's administrator see I'm downloading the video?
I can't find much information about that specific add-on on-line other than its page for download, but some educated guesses (with oversimplification, I'm not sure if you're asking this question from a technical or privacy standpoint, but I'm assuming the latter):
The add-on will download it presumably in a similar way to the way your browser is downloading it anyway. That is to say, if you go to youtube and view a video, what actually happens is that the video is downloaded and temporarily stored on your computer (in chunks), so presumably the add-on simply does the same thing but makes sure it's permanent. This means from a privacy standpoint, you're basically in the same place as if you simply navigated to the youtube page.
There is absolutely no reason why any third party is necessary, that is to say, there is no reason for the add-on to have www.examplebadsite.com handle your downloading, you can pull it from youtube directly just fine, with no interaction from ANYBODY else.
In a nutshell, youtube.com (etc.) can know what you are doing, but they will not likely care. The creators of the add-on can know what you're doing, but they don't need to know. The software to download the video is already on your system, after all, you download the video to view it on the youtube site. That said, the specific creators of this specific add-on could in theory have bad motives, and track everything you do, I cannot vouch that they don't - I highly doubt it though.
Typically, these kinds of video downloading tools just figure out the URL to download the video from, and download it to a file. Youtube would not be able to tell the difference between you viewing it online, and downloading it.
There is a possibility that this particular add-on will collect info on your activity and upload it back to their server, but I do not think they would do so. People generally make it known when they realize that something is phoning home like that.