We are running Joomla v 2.5.6 and have the JLinked Share module installed, version 2.2.
It appears that as of late the LinkedIn share button is not working in Chrome. Upon clearing the browser cache it works the first time, however if I try it again on that same page the share window does not appear.
This issue does not appear to be happening in IE or Firefox.
I'm not sure why the LinkedIn Share button isn't showing for you. I just tested with Chrome v33.0.1750.152 on a Mac and the Share button shows just fine after multiple reloads and waiting about 10 minutes between some tests.
Can you recreate this on other computers? Can you check what Chrome extensions you have installed as well and try disabling them? It's possible one of your add-ons is causing the conflict.