How can a passenger tell if a flame-out / engine failure has occurred?
Mainly by the soon to happen descent and landing, probably at a place you weren't expecting to go.  :)

If you are seated behind the wing, you might be able to see smoke or flames coming out of the engine, but other times it will just quit with no visible indicaion, depending on the cause of the failure.

Sometimes (usually just after takeoff when the airplane is slow) it is obvious and you can tell by the change in the sound of the airplane (one side suddenly gets quiet or there might be a loud bang depending on why it failed), a sudden yaw in one direction (the nose pulls to one side), and an abrupt pitching down of the nose (done by the pilot to maintain flying speed).

Other times (like during a descent with plenty of speed and the engines at idle, or when the center engine of a three engine airplane fails) it can be hard for even the pilot to tell without looking at the instrumentation, so you probably won't notice it. 

Another major clue would be when the seatbelt sign unexpectedly coming on, without an announcement or indication of turbulence, and the flight attendants suddenly putting things away and getting ready for landing, especially when you aren't close to your destination. Being briefed for an emergency landing is also a dead giveaway (no pun intended).