I wrote my reply to your post on Twitter on my site and syndicated the response to Twitter so you would see it and get the notification. The “screenshot” you mention is actually a blockquote embed of your post/thread, so that I’ll have a reply context on my site so that anyone reading it there (or in a social reader) will have some context to understand my reply.
You’ll notice that your response is sent back to my original and appears as a comment there.
Now I can respond to that (again on my own website) and easily syndicate it back to the original thread for you to see.
We could easily do this conversation back and forth on our own websites using some of the building blocks of the IndieWeb (many of them W3C or web recommendations or specifications). Since you choose to use Twitter for communicating and as part of your online identity, I’ve set my WordPress site up to accommodate your choice.
Fortunately the open web has some easy methods for us to communicate without silos like Twitter, Facebook, Instagram, etc. intervening. It may require you a bit of work to set your WordPress site up with a few plugins to do it, but there are also a handful of inexpensive services you could use to link your domain name of choice up to and let them handle the administrative overhead and management of your site if you chose. If you were ever unhappy with them, you could export your content and take it along with your URL to another service or manage it yourself. Ultimately, competition to provide the best service will help rapidly improve the ecosystem.
The IndieWeb is simply a group of people trying to make all of this easier and better since the corporate social media companies aren’t interested in solving our problems.