A remote is just a clone which you have some way to access. Commonly it's accessed via the git protocol or http (for reading) and ssh or https (for writing), but filesystem access works just as well. All you need to do is clone your repo:
# clone project into project2
git clone project project2
The origin remote will automatically be set up in project2, pointing to project, just like you're used to. You could add a corresponding remote in project, pointing to project2, and try out whatever fetching and pulling you want to do.
If you want to try out pushing, make sure you make a bare clone to push into:
# the .git suffix is traditional for bare repos, since they're
# basically the .git directory of a normal repo
git clone --bare project project.git
cd project
git remote add bare-remote ../project.git
# now you've got a remote to push to
I'm not sure why you're naming them "local_brach" and "remote_brach" (is that meant to say "branch"?). These aren't branches. They're clones; they can have their own full sets of branches, probably with some corresponding.