Voltage drop across emitter of BJT
Okay, I think somewhere in here one of these matches your verbal description. I ran the simulation with the built-in circuit simulator and the voltages are correct for the type of LED used. 







simulate this circuit &ndash; Schematic created using CircuitLab



R7 just simulates the input resistance of your meter, so the emitter is open. 

In cases 1 and 3 the transistor is saturated and has &lt; 0.1V across it (collector to emitter).  

In case 2 the transistor has about 0.5V across it (less than the usually quoted 0.6 or 0.7V because the meter draws almost no current). 

I don't see anything that resembles 1.2V there. If it's an IR LED then you could get 1.2V and then V1 would be 1.2V but the other voltages should be the same as shown above.