It all depends on what you want to do. If you are looking at 10,000 uploads a month and each upload is on average 5mb per file (decent quality .avi or .mpeg file that is about 1-2 min long) you are looking at 50,000MB storage. Now during the upload process you have used 50,000mb of your bandwith, this excludes the loading of the webpages. Now let say that 5,000 people watch only 1 video a month, that is another 25,000mb bandwith. Also let's say 1000 people decide to download the clip that is another 5,000mb....see where I am going with this? Each time someone uploads/downloads/stream data it count towards your bandwith. There is also storage which is huge, unless you cap peoples storage.
So to answer your question; yes a dedicated server will probably be best.
Another approach might be to create an API with another service like photobucket. Store all you photos and videos there, then the only disk space required on your webserver will be the webpage itself and the video uploading/streaming/downloading will not count towards your bandwith, cause photobucket will deal with that.