My wife is concerned some of the information on Gramps will wind up on the Internet. Everything I have read, so far, says the data is stored in the local .gramps
file and nowhere else.
Is this correct?
Gramps does not publish to the internet within itself.
Gramps stores data locally within a SQLite db file. One file for each tree that your installation is set to manage.
The .gramps file is a backup. It is not the live tree data. You may also have a .gpkg file which is a newer backup format.
When you attach media to an individual in your tree, photo, certificate, etc. The files hold a link to that media file, the media isn't held within the gramps database itself. You could make Gramps link to a file held in the cloud, however in this case Gramps is reading the internet, not publishing to it.
If you wish, the .gramps file is compressed XML. You could decompress and examine it to fully set your mind at rest.
See the Gramps Project Wiki for more details.