All of that changed with the new livejasmin.com interface. On the left side, you will see a vertical series of button. The last icon is the one to switch on your webcam. Then, the white adobe flash message should appear & you should click on allow.
Only after clicking allow, your webcam is going to lit on. If you have a problem with your webcam while you are in a private chat, i recommend to do a right click, click on settings and then click on deny and allow to re activate the webcam.
Also, you may click on the first icon from the message box above (the one looking like a screen) and then untick the option: “Enable Hardware acceleration”