Hidden wire detector, using resistor
Having built numerous multi-stage bipolar amplifiers as a kid, I'd wager this resistor is to limit current.

Those amplifiers, lo those many years ago, would produce a power-line humming or evil-singing/buzzing (from quick rectifier diode turnon) as I brought my hand near the floating input of my breadboard multi-stage audio-band gain-of-100,000X circuit. There was enough gain the ELECTRIC FIELDS were being detected.

Thus very similar to this 3-transistor very-high-Rinput circuit.

Yes, I'd include resistors in all three collectors. Perhaps 1,000/1,000/100 ohms. 

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Remember the input lead is floating, your body is the return path for the induced charges as you hold the circuit near the wall, as you watch for a slightly-glowing LED.