In Social Media platforms, like Facebook or Instagram, which mechanisms/tools you use, beside direct surveys, to study if those who follow you are also your loyal clients?

If they are able to share what they see from my platform

You should give away some discount codes (one for Facebook, an other one for Instagram). This way you can measure the amount of people who use the codes while purchasing your products. Furthermore you can share some interactive posts like "Show off your new "product" and use hashtag #product to have a chance to win whatever..." You can see who purchased your product and posted online with your hashtag.If you are not exactly a seller, or you rather provide a service you can hire some hostess girls who go around in the venue place with an ipad to ask the clients kindly to press like to your page or follow your account. If they do so, they get some presents, discounts, or opportunity to win something. This way you can make sure that the people who are on your page are your loyal customers.

Good question. Review section (in most cases) shows opinion of loyal customers. Movement on social media platforms after on-ground actions like promotion, in-store activity can be treated as loyalty.