one party has to have an internet connection,
eventually.you can trade signed transactions around like a file for a long time and then someone else goes to the internet cafe to bulk upload eventually.
no different than the practice of bundling credit card transactions together to reduce static fees, or settling a bunch of checks at the end of some time interval. it is a temporary trusted environment in all of those systems including bitcoin, no big deal. keep track of anybody whose payment failed.
bitcoin transactions can be signed and traded where only one party has an internet connection.
bitcoin transactions can be signed and traded where neither party has an internet connection.
it just has to be broadcast and added to the chain eventually. so even in a world without ubiquitous internet this concept doesn't go away, it will always exist in some form or anybody can recreate it or perpetuate a blockchain that has fallen into disuse.