You can do a lot with the setup you already have: work on your Puppet manifests as a team, communicate changes via a central Git repository, and manually apply them on a machine using the papply script.
However, you still have to log into each machine to update the Git repo and rerun Puppet. It would be helpful to have each machine update itself and apply any changes automatically. Then all you need to do is to push a change to the repo, and it will go out to all your machines within a certain time.
The simplest way to do this is with a cron job that pulls updates from the repo at regular intervals and then runs Puppet if anything has changed.
You'll need the Git repo we set up in the Managing your manifests with Git and Creating a decentralized Puppet architecture recipes, and the papply script from the Writing a papply script recipe. You'll need to apply the bootstrap.pp manifest we created to install ssh keys to download the latest repository.