Yes, we do share our job openings on our LinkedIn.  Most of the time, the team only shares the jobs they are actively working on (otherwise it's too many).  We encourage our clients to share their jobs that we are helping recruit on their social media. That way they get in front of others that know them and may make a recommendation.

I will say when I share one of our internal openings on my page, I get great results from others sharing it along and folks contacting me about the opening. I'd definitely recommend it.

I've used my personal Link'dIn to share a video asking my fellow HR peeps in the financial industry to share information about an opening we had. While I didn't get any new applications as a result of the video, I got some great feedback from folks who thought the video was a great idea and it definitely got my CU some great exposure. I have a very small following right now so I think if I nurture my account a little more that I would have greater and more positive results. 

We did this a lot at my prior company. We even did a little bio on the hiring manager to show why someone would want to work for a great leader like them. Without having numbers, I believe it was really effective and engaging. Best of luck!