As you say, much of the data one posts may have little value and feel ephemeral, but certainly not all of it, and certainly not in aggregate. At least the individual should get to decide and have agency over the decision. As it stands, I can delete individual posts from Facebook, but I have no guarantee that the data is physically removed from their servers and still available for either their internal use or for possible future governmental use.
Another way to frame it all is to think of your web presence as a commonplace book.
If you recall the early days of social media, you may appreciate this alternate viewpoint of social media that I wrote about a few months ago: http://boffosocko.com/2017/04/11/a-new-way-to-know-and-master-your-social-media-flow/
Interestingly, I came across your post almost immediately after fleshing out some detail on the wikipage for cross-posting which may be a worthwhile overview from the perspective of a traditional social media user. To help conglomerate all of the various pieces for you and others in the future, I’ve created a category page under the heading “syndication” with links to all of the various pieces which may together make a more coherent whole.
As for your question (excuse my rough translation):
Then I think again, if I put my tweets first on my own site, what about the possible conversations that result from it? If someone answers and I reply again, do I do that on my own site? The IndieWeb wiki is not very clear here…
There isn’t a direct answer within some of the pages you mention, but ideally, yes, all of the conversation takes place in a back and forth manner on your own website (as well as that of those with whom you’re communicating). Sadly, not all of the moving pieces have been solved completely with respect to user interface which could be done in multiple ways. One standard in particular that isn’t supported by many is that of salmention. Until then, some of us are managing to do this manually to maintain the threaded comments so that the entire context of a conversation is still available on our own sites. Even without it, some semblance of threading is possible by providing permalink URLs for all the parts of the conversations on individual pages until such time as it’s more feasible. If you care to experiment, try commenting on this on my site and see what happens.
Incidentally, especially if you haven’t come across it yet, I hope that as you continue to explore and write that you’ll syndicate your content to https://news.indieweb.org/nl for the benefit of others.