Does rain (temporarily) deplete the surrounding atmosphere of carbon dioxide?
Here is the mass-calculation. We will consider a column of the atmosphere with a footprint of 1m × 1m. This column weighs about 10,000 kg (per square metre). In these days of climate change we will assume the current average CO2 concentration is 400 ppm, yielding a total mass of CO2 in this column of 4 kg. 
The rain doesn't wash out the entire thickness of atmosphere but only (in general) the lowermost troposphere. We will assume, generously, that some 2.5 kg of this 4 kg of CO2 is available to be scavenged by raindrops. 
Now consider a cloudburst of raindrops, equivalent to some 100 mm  on the ground. So we have 0.1 cubic metres, or 100 kg, of fallen raindrops within our square metre column of air. The solubility of CO2 in air is strongly temperature dependent, up to 2.5 g per kg at 10 deg C. But at such a low temperature we couldn't achieve 100 mm of rainfall, so we will compromise at about 1.5 g per kg at 25 °C. So, assuming complete CO2 saturation within the raindrops (unlikely) our 100 kg of rain will contain 0.15 kg of CO2 , scavenged from about 2.5 kg of CO2 in the 'rained out' column of air, or about 6% of the low atmospheric CO2 . 

Bear in mind that this assumes an enormous rainfall intensity, 100% CO2 saturation of the water and equilibrium chemical dynamics. After the raindrops hit the ground at least half of it will immediately re-evaporate back into the air, leaving, at absolute most, about 3% of the atmospheric CO2 leached out of the atmosphere that will be available to react with the soil, rock or biosphere. Also consider that this is but one of several important processes affecting CO2 transience, such as photosynthesis, respiration, volcanism, industrial pollution, etc. So the CO2 estimates that you read about are average values. Advection and turbulent air mixing should ensure that the CO2 regains approximately normal concentration within an hour or two after rainfall.