I'm currently at COP24 (United Nations Climate Change Conference) in Katowice, Poland.
(some editing since I took a screenshot of my laptop)
I've asked some guys about $\ce{CO2}$ emissions but very few have sufficient technical knowledge.
My assumption is that they assume the data is right.
Some Google-fu:
More recent energy statistics are sourced from the UN Statistical Office, which compiles data from official national statistical publications and annual questionnaires.
https://www.thenakedscientists.com/articles/questions/how-do-you-measure-carbon-dioxide-emissions
The Intergovernmental Panel on Climate Change has published a five-volume set of guidelines that all the countries now use, as part of the UN Framework Convention on climate change, for estimating emissions on all greenhouse gases and it does produce uniformity across countries.
Although there are some large power plants in which they actually put measurement devices in the smoke stack and can measure the amount of CO2 that comes out, that is unusual.
About measuring $\ce{CO2}$ in the atmosphere, we have a satellite for that: https://climate.nasa.gov/climate_resources/99/graphic-measuring-carbon-dioxide-from-space/
About air quality index - https://en.wikipedia.org/wiki/Air_quality_index - we have a number of standards for that (shame there are multiple standards)