I can’t figure out what the etiquette is when driving around the square.

Here is my guess. There are clearly two lanes. Yet most people do not use them independently. It’s more like one large lane and you should only pass another car that is not moving. This is polite small town driving.

There is however the lunch time exception*: during lunch time there are two lanes. One lane for those who have a limited time for lunch and one for the rest. The most critical aspect to this exception: DO NOT dilly dally in the “lane of limited time”.

*Driving experience in LA and/or New York City helpful during the lunch time exception.