This depends on what you want users to be able to share...
If you just want users to be able to say something about your institution, then many social media sites allow you to make direct links to share your specific website. For instance, on Facebook, you could use a link like:
Anyone clicking on that link is invited to make a post about "www.mysite.com". There are similar links for Twitter, Reddit, and Pinterest; I've never played with WhatsApp but I would guess they also have something similar. These links could easily be added to the side menu in the app using the Moodle Mobile additional features plugin.
Now, if you're looking to share a particular course, or content within a course, that's another matter, and would likely go into custom development territory.
Hope this helps,