This isn't the best design, but if you want to share what you're reading on Project Gutenberg, you can. Right after you download the ebook, go to the social page and you'll be offered the option of tweeting or posting a Facebook update. The tweet will look something like this:
Project Gutenberg also has a Twitter account and RSS feed where they post mentions of newly converted (or recently updated) public domain ebooks which have been added to the collection. You can find those on the social page as well.