The part that makes a party line work is the line driver, which is a voltage-controlled current source rather than a voltage amplifier. RTS call theirs bilateral, meaning it can both source and sink, which is what lets one pair carry everybody in both directions at once.
Because each station injects a current rather than asserting a voltage, the signals just add. They're converted to a voltage once, at the single termination resistor, which is why there's exactly one termination in a system and why it matters so much. A current source can't short out its neighbours the way a voltage source would, so stations can be added anywhere along the line without any of them fighting.
The numbers behind it, from RTS's handbook: the current source has an output impedance of 10 kΩ or greater and a transconductance of about 3.3 millisiemens (5 mA out per 1.5 V in). The microphone preamplifier ahead of it reaches 54 dB of gain, and an electronic switch sits between the two so that a closed talk switch disconnects the preamp from the line entirely rather than just muting it. That's a noise decision: sixteen idle preamps all connected to one pair would audibly hiss.
On the listening side, a headphone amplifier of 30–40 dB drives a 50 Ω headset to around 105 dB peak, which is what a concert or a sports floor needs, with the headset's own isolation doing the rest.
The sidetone null is a differential amplifier, and that's why it drifts. The open-source ComClone beltpack does it the textbook way: the local microphone signal goes to one input of an op-amp and the audio coming back off the bus goes to the other, with a trimmer setting how much of your own voice is subtracted from what you hear. The null is therefore a balance between your own signal and the line's response, so it holds only for the line it was set against. Change the cable length, the pack count or the termination and the balance moves, which is the same behaviour people report from hybrids further up the system.
Worth noting that not every party-line station is built the way RTS describe. The handbook says these systems use current sources "or similar electronics", and ComClone shows what the hedge covers: it couples its audio onto the bus through a capacitor rather than a current source, and interoperates on a Clear-Com-style line regardless. The current source is the reason the topology scales: a simpler driver works fine until enough of them are sharing a line.