I don't know about twitter or linkedin but for facebook :
Currently, posts can share on facebook profiles with facebook share buttons (with many theme's social share buttons under the posts)
For facebook group or facebook page publishing;
You can take a look at Facebook Connect. They got widgets, but job is not finishing on BlogEngine side, you need to create an Facebook aplication and settings etc.(maybe you already know)
In my experience ; i able to publish posts on a Facebook Group with facebook connect, but it required some coding, i think create a facebook page much more useful than create a group.