LinkedIn share content option like facebook request dialog
No, currently there is no way to share to the individual persons using LinkedIn API or widgets. Only two options are available for share API. One of anyone: all members or connections-only: connections only.
Widget has only url, title, source and summary parameters. So even if you get the list of the connections via API and create your own dialog with checkboxes you still can't share via API or via dialog box to individuals.
See LinkedIn related feature request