That is definitely not the case 
GitHub is just a public place to host git repo’s. What is on your machine is simply a copy of a git repo. If you’re the only one with write access to the git repo you push up to Github, then only you can make changes in there. Other’s can submit “Pull Requests” (PRs) to suggest changes but you have the say on whether you accept those or not.
As for your machine, all activity on the git repo on your machine is controlled locally, either edits, or “pulls” from the gitHub repo if you choose to accept PRs.
[EDIT] It’s worth adding that you don’t have to use GitHub to use git for version control. I have numerous git repo’s that are only on my computer.
Nah, its first thing in the morning!