Ok, I just discovered the fix. In order to copy to a network dest (UNC), you must first delete the dest file if it already exists.
Delete File( "\\mywork2.work.com\folder2\dest\test1.csv" );
Copy File(
"\\mywork1.work.com\folder1\source\test1.csv",
"\\mywork2.work.com\folder2\dest\test1.csv"
);
If you don't first delete out any existing file, presumably as it has to overwrite it, it will not execute the copy. I don't know if it's possible in the future to put a switch or variable in the Copy File() like overwrite ("Y" or "N"). That would make this a little more obvious probably.