You don't need to (and should not) use :: or ::: for function of you own package.
A function is not exported to user is not present in the NAMESPACE. If you use roxygen, putting @export tag ONLY for the function you want to export will do the job.
If you want no Rd file, @noRd is the way to go. However, if you want to produce a help file but not list it in the index you can use @keyword Internal. it will still be accessible using ?my_function
As a package as internaly access to its own namespace, every functions is available to use, not just the exported one. You don't use :: or ::: for function of your package. Only :: for functions exported in other package. You got the NOTE because CRAN discourage the use of an unexported function of another package.
Hope it helps