Who said anything about a hosting server? Why isn't there a simple option to export a conversation to local storage, encrypted or unencrypted, along with a warning that 'your conversation is now leaving the secure Signal zone.'
This is one of the problems with Signal having a bit of confusion about what exactly it's use-case is. There are plenty of cases where locking down the ability to save/view/export messages are valuable, and Signal provides tools to be able to do that. Making that the mandatory case though means that it's harder to adopt as a general-purpose communication platform.
The need to decide if the goal is still to get as many people off of SMS/facebook-messenger as possible, or if the goal is to provide extreme security to dissidents and protestors, or if they're going to spend the effort to be able to do both effectively and let you choose which conversations or messages get which level of protection.
It's like when Signal asks you to put in your PIN, but it's the same PIN you use to unlock your phone. There's a different Signal PIN, but that functions as a check for when you install Signal on a new device. Your regular PIN is just a repeat of your phone PIN, and thus adds exactly zero security.
Meanwhile if my counterparty and I have communications in Signal that we want to preserve at scale, it's impossible to do so. A simple example would be that I have years' worth of conversations with my wife that I can't easily back up any where. We could export every picture and video by hand and screenshot or copy the text of every message, but that would be extremely time-consuming and tedious.
That's the basic problem. The data is already only semi-secure in that it's subject to exfiltration without consent. It's just inconvenient. And where parties do consent, it's very inconvenient because it's many many hours of work; in the end, it's just more security theater.