Google can index pages with no title page. When there is no title tag, they will choose a snippet to use as the title.
Google does recommend using a title tag as an SEO signal that will help your site rank. It might be better to choose your own snippet from the post. 3 to 7 words with the author name and the site name in the title is probably going to be better than what Google comes up with.
I have never tested trying to share content with no titles on social networks. My guess is that, like Google, they can do it, but it won't look as good as if there were a title.
Taking a step back, why would each short post need its own full page? Even if searchers do land on your page, I doubt they will have a good user experience. Landing pages need enough content to satisfy the user. That is usually a lot more than just a sentence or two. You should think about using this content to build pages with multiple posts on each page rather than a single page for each post.