According to the Ultimate FAQ Sony published on the PlayStation Blog, the PlayStation 4 will not allow users to share videos via YouTube or copy recorded footage to other devices via HDMI cable. However, this is a limitation of the console at launch. These two questions do elaborate to keep following them for updates after launch for when it will be available.
In the meantime the FAQ goes on to explain how the share button ill work and where the videos can be uploaded to. The PlayStation 4 is constantly capturing video, but only keeps the last 15 minutes of gameplay, and then upon pressing the share button the PS4 will take that footage and let you edit it with the Share button interface (presently you can only choose a starting and ending points), upload it to Facebook and screenshots to either Facebook or Twitter. You cannot add voiceover to shared clips, but you can record speech through the headset while playing.
Users will also be able to stream via Twitch or Ustream, but for the streaming to be archived you will have to specify it with the Twitch (Ustream just wont) from your account online.
The PlayStation 4 will launch November 15th in North America and November 29th in Europe.