Which Image Gets Included When You Share a Page?
When someone shares your WordPress page on Facebook or Twitter, an image or video usually appears as part of the post or tweet. You can improve the chances of the right images being included, but you don't have full control.
Here's an article about how social image sharing works from Yoast.com.
For a congregational website in WordPress, you can use the Yoast SEO plugin, which helps with how page content is displayed in search and social sites. Here are instructions for customizing for sharing on Facebook and on Twitter.