Speculation
In the US, the AC mains has a "live wire" and a "neutral wire". One wire is kept near 0V while the other has 120 V RMS (meaning it swings from +170V to -170V). I think that the phenomenon you see takes place at the "live wire" only. That is - there is a local rapidly varying electric field at the tip of the live wire, and that is responsible for initiating the luminescence you are seeing.
You could confirm this if you had the fortitude to do this experiment with the power used for an electric dryer - there, the two wires are both live (that's how you get 240 V... it's necessary to get really high power devices working in the US where the normal mains voltage is 120 V). If my theory is correct, you would see both ends glowing.
Caution! Potentially lethal shock hazard!
Because of the use of exposed bare wires in this demonstration, it should be done with great care and capable supervision.
Electric pickle is an extremely dangerous experiment. It will get a lot more dangerous when you double the voltage. It might be better to use an AC/AC transformer with a floating secondary - then the fields ought to distribute evenly without stepping up the voltage. Even at 120 V it's a dangerous experiment.
UPDATE
There was another question (since deleted) on this topic, linking to a video that showed that the pickle could sometimes light at one end, and sometimes at the other end. Looking closely at that video I now speculate as follows:
- The light was coming from the place where the metal touched the pickle, and looked like "fire". This could be spontaneous combustion of the hydrogen and oxygen gas that are produced from hydrolysis at the electrode (hydrogen during one half of the AC cycle, and oxygen during the other half). The burning would then create a local hot spot sufficient to cause Na+ ions to luminesce.
- It seemed to me that the light was coming from the side where the fork was inserted less deeply: this would encourage higher current density and thus heat; once one side "catches fire" I imagine the combustion product (steam) could increase the local resistance, so more power would be dissipated at one end - in other words, it might be an unstable equilibrium.
- The one experiment in which first one side, and then the other side, lit up, may have been a result of having the cut (high resistance) in the middle: this made the voltage distribution a bit more even (the moment one side became more resistive, the total current would drop, the voltage drop across the middle cut would be less, and that would "dampen" the effect that normally drives luminescence to one side).