I've heard from some readers asking about possible limitations to the data. The CDC's MMWR (detailing older figures than we reported) spells out the role of fragmented reporting systems, though it's not clear how widespread this has been, nor what overall impact it may have had.
The CDC defines "reporting entities" as jurisdictions, territories & fed entities — so to the extent that there may be double counting, it likely involves crossing state lines for a second shot, or, say, getting one dose from a tribal health clinic and another from a state site.
While other caveats about data quality mapped out in this MMWR do apply, there doesn't appear to be any large scale problem with counting, say, people who got Dose 1 from a Walgreens and Dose 2 from a CVS in the same state, state officials told me. https://www.cdc.gov/mmwr/volumes/70/wr/mm7011e2.htm
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