I have a slightly different idea, I can’t bother to write the code right now (starting at 2am), but gave me a bad concept:
(I assume you are using JqueryUI.autocomplete)
so you need to catch the create event (look at the documentation for the category section in JqueryUI autocomplete to understand this).
So, on the create event, you want to add two divs inside the div, which are autocomplete bar / window / whatever and float them with a width: 50% ;. then when you retrieve the values (via ajax), check the “category” (that is, listLeft or listRight) and then use .append to add to each list accordingly and make sure you give them a Button autocomplete class or its callable.
graphical representation:

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