The incident happened on Monday afternoon on 200 Market Street, not far from the Fourth of July festival held in the city.
Police said a 20-year-old man was shot once in the stomach and once in the right leg.
He was transported by doctors to Central City of Jefferson, where he was listed in critical but stable condition.
Officials say the incident began as a verbal argument between people in two vehicles that ended at the intersection of 2nd Street.
According to police, the driver of one of the vehicles threw a drink at another vehicle. That resulted in bottles being thrown back and forth between vehicles.
As the two men got out of their car, police said the man who had thrown the first drink opened fire.
Police were alerted to the area when the shooting occurred.
Officers were then able to quickly arrest the two men involved, pending investigation.
A weapon was also recovered from the scene.
Police say that two individuals have permits to carry.
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