Back before the net was made availible to the public people used BBS's and various networks such as FIDO net and VBBSnet. It was slower and somebody had to foot the long distance bills. Back then there was no such thing as free long distance. You paid through the nose to call somebody sometimes a block away from you but in a different zone.
Today using the telecomunications backbone or at worst the free long distance that is common a private network could easily be setup that duplicated the function of the Internet. If you had to resort to modems high bandwidth items like streaming video would overload the system but email, websites and such would function quite well.
Community TV was a growing medium before the internet. The net soon made it obsolete. It was far more expensive, less responsive and required expensive hardware.
Print medium such as letters and newspapers are still used heavily.
Texting is about to blossem into it's own medium really. Right now it's still rather primitive but it's growing fast in capablility and usage.
Voice. Given todays technology it's no longer necessary to speak real time. Voice mail tag can carry on extended conversations. There is of course the one on one system.
What all these lack except the private network is the ability to speak to millions around the world at the same time or in non-real time fashion.