I’m using the twitter gem, but I’m okay with API answers. With google analytics, I have access to shared URLS and their effectiveness:

I was wondering if it’s possible to track down which user shared which link?
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You will have to use the search API – but there’s a catch.
Twitter’s search API doesn’t index the t.co URL – so you can’t search for t.co/0uxMcDrD.
However, Twitter will allow you to search for the URL which is wrapped up in the t.co shortner.
In your case, all those URLs point at picatic.com
So, a search like https://twitter.com/search?q=picatic.com will reveal every tweet which links to your site and is, therefore, wrapped in t.co.
You can then see which users have tweeted about your site and linked to it.