Your question intrigued me so I did some research.

First of all, there is always an option to just download the files in
Synaptic, so one could save them that way. But where are they? On this
computer (running Linux Mint 6, but it seems to be the same on my Ubuntu
system) I found them at /var/cache/apt/archives. The names are somewhat
cryptic, so make careful note of what you are installing! I downloaded
Backgammon, which goes by gnubg.

In that same folder were several hundred other files, which I think are all
the other software I've ever installed, including lots of library files.
When I installed the Backgammon game by clicking on the main executable it
informed me that another file (containing data) would also be installed -
very handy.

Note that no matter how much you copy over from /var and /share you still
have to install the main .deb files on your other system. Many mysterious
changes are always required which only the installation can make and we
really don't need to know about. Fortunately the installer is easy to use
and works flawlessly as long as all the library files are there.