How to convert Sodium Carbonate to Sodium Bicarbonate at home (CO2 scrubber)
A process commonly used to teach $\ce{CO2}$ scrubbing to chemical engineering students involves not sodium bicarbonate, but sodium hydroxide ($\ce{NaOH}$).

The reaction is exothermic and is as follows:

$$\ce{CO2 (g) + 2 NaOH (aq) -&gt; Na2CO3 (aq) + H2O (l)}$$

The aqueous sodium carbonate is subsequently treated with calcium hydroxide to precipitate out calcium carbonate:

$$\ce{Na2CO3 (aq) + Ca(OH)2 -&gt; 2NaOH (aq) + CaCO3 (s)}$$

Disclaimer: 

This is usually done with much higher concentrations of $\ce{CO2}$ (e.g. >10%) than you would find in the atmosphere (~400 ppm). The point of scrubbing in a chemical plant is to reduce emissions to a level that is still much higher than atmospheric $\ce{CO2}$ levels but lower than governmental standards. I suspect that you would not see a noticeable change in ambient $\ce{CO2}$ levels from putting something like this in a place with such (relatively) low $\ce{CO2}$ to begin with. There is a very low driving force to push gaseous $\ce{CO2}$ into the liquid and solvate it when the partial pressure of $\ce{CO2}$ is low.

There exists a unique relationship between the partial pressure of $\ce{CO2}$ above a solution, the dissolved carbon in the solution, and the pH of the solution, shown here (own work):


Note that the equations on the right-hand side are in solution.

So if you wanted to see what was going on in a controlled environment with no other species in the solution / that could enter the solution, you could just use a pH probe to see how much carbon was pushed into the solution!