Ask Mr. Smarty Plants

Salvia coccinea (Scarlet sage), according to the USDA Plants Database is native to the lower 48 states of the United States, occurring in Texas, Louisiana, Alabama, Mississippi, Georgia, Florida, South Carolina and (strangely enough) Ohio, as well as Puerto Rico and the Virgin Islands.  The USDA PLANTS Database says:

"In PLANTS, native means naturally occurring at the time of Columbus. Introduced plants arrived later, invariably with human assistance, from some other part of the world. Introduced plants reproduce spontaneously in the wild without human help. ... Because people have been moving plants for thousands of years, and because it is often hard to know how a plant got where it is, Native Status is frequently ambiguous and therefore difficult to assign.

We have broken the PLANTS Floristic Area (PFA) into Native Status jurisdictions to try to improve our information about where plants are native. A plant that is native to any part of a Native Status jurisdiction (e.g., L48, the lower 48 states) is considered Native, even if some populations within that area are introduced."

The Atlas of Florida Vascular Plants also lists Salvia coccinea as native.

The plant is also native to Mexico and Central and South America.  What Wikipedia says is: 

"At one time Brazil was considered to be where it originated, but its diploid chromosome count now points to Mexico as its place of origin."  

This doesn't tell us when it originated and how it spread, but it is likely that it originated a very long time ago and could have spread up into Texas and towards Florida long before Columbus came to the New World.  As the  USDA PLANTS Database says:  "Native status is frequently ambiguous and therefore difficult to assign."

Escaped cultivated plants likely explain the population in Ohio.

You can read more about what USDA PLANTS Database means by "native" and "introduced".