Not necessarily, they might just be radiating humid air. It's the evaporation itself that leads to the cooling.

But the humid air makes it harder for people around you to cool evaporatively (eg. sweating)

Well yes, you must (irreversibly) increase entropy somehow. The air itself won't be warmer though.

This is why we have wet and dry bulb temperatures.

Your intuition about temperature is focused on the dry-bulb temperature that a simple probe gives you.

A wet-bulb is literally a probe with a wet cloth or tissue around it.

As the humidity in the air increases the wet-bulb temperature approaches the dry bulb temperature.

This is absolutely “warming”. The latent heat of the air increases. The sensible heat remains the same.

Without the jacket the person will emit the same heat. The jacket fans do generate a trivial amount of heat, but it’s likely less than a desktop fan.

ventilation is a lot cheaper than AC though

Hey as long as I feel cool and look cool I don’t care what it does.