I am going to make some guesses here, so please correct me if anything is incorrect...
Most Chocolatey Packages that exist on the Chocolatey Community Feed are wrappers around the native installers, that exist on the internet, and which are downloaded at installation time. These installer packages are placed onto the first machine, so that the installation can happen, however, Chocolatey is also quite good at tidying up after itself (where it can) to minimise the amount of space taken up on the hard drive.
As a result, what you are describing likely won't be possible, without introducing the concept of package internalization. This is described in detail here, an important part being this:
NOTE: Many repositories have a concept of a proxy repository. Unlike NuGet repositories, you likely DO NOT WANT a proxied NuGet/Chocolatey repository pointing to the community repository. They only cache packages - cached is not the same concept as internalized. To reuse packages from the community repository in a reliable way, you need to internalize them. The community repository is subject to distribution rights, which means many packages need to download things from the internet at runtime. That's unreliable and a no go for many organizations. You can use Package Internalizer or manually internalize packages you want to use from the community repository.
If you go through the process of internalizing the packages, then any computer on your network, that doesn't have internet access, will be able to consume, and install, those packages.