Short wave communication using digital techniques is not necessary for simple communication; voice and CW (Morse code) work just fine. Where digital communications becomes the best (or only practical) solution is when one site has in it's possession, say, a medical textbook. Large amounts of text or graphics cannot be transmitted in any other way. I could list many more examples. In my opinion, attempting to reconstruct something like the internet isn't necessary, nor is it practical for many years after the post office.
I think communities will form a system something like the old telegraph office, where an individual wishing to communicate with, say, a loved one in another community will go to the "telegraph (radio)" office and fill out a form indicating what they wish to say. If voice communications can be established between the sites, a system could be worked out much like the current MARS (Military Affiliate Radio System) system that allows service personnel to talk on a radio at their end while a ham radio operator stateside connects his radio to a telephone (using a device called a "phone patch") and calls the stateside phone number collect. This allows individuals to schedule a time when they can actually speak to someone at another site.
PC to PC communications using HF radio is slow (digital HF communication is limited to 300 baud, VHF to 9600 baud, UHF can have very high baud rates, all constrained by bandwidth considerations) compared to what we are used to on the internet. As I stated above, I do not envision this as an internet substitute where everyone has a PC and radio; but as a community resource just like the power generation station.
Offered by Ron.