You should use the command git daemon to host your repo, like this:
In your computer that will act as a server:
git daemon --base-path=<path_to_folder_containing_project_folder> --export-all
(please note that path_to_folder_containing_project is the folder containing your projects folders, it will provide all projects under that folder)
In your client:
git clone git://<local ip>/<project name>
The cloned repo will have its origin pointing to the server in your LAN, so you may want to use git remote set-url origin to point it to the original origin.
You may want to run git daemon with the --verbose option to get more details in case you run into problems.