In the USDA’s Plant Profile map for Cerastium arvense, it shows the species occurring across North America and as both native and introduced in most states, provinces, and territories. Why?
The answer is that field chickweed is a species that has taxonomically-disputed subspecies and varieties in both North America and Europe. Following Flora of North America’s treatment, two subspecies in North America are recognized: the native (and generally larger in most dimensions) Cerastium arvense subsp. strictum and the European-origin Cerastium arvense subsp. arvense. This is why the species is listed as both native and introduced throughout much of North America, as both of these subspecies can co-occur in an area. Even when the two subspecies overlap, though, they are not interfertile and so maintain their (subtle) distinctiveness. However, if you look at the FNA page for subspecies strictum, you will see a long list of synonyms suggesting that many taxonomists have had their own interpretations; even the writer admits it is challenging to decipher the group:
Subspecies strictum is widely distributed and grows in a great diversity of habitats, making it difficult to circumscribe and distinguish, both from subsp. arvense and from forms of Cerastium beeringianum, C. velutinum, and C. viride.
An enterprising researcher who likes a challenge will perhaps tackle the group one day.