Dangers of getting replacing iPhone 6S screen replaced at third party store I'm considering having my 6S screen replaced at a non-apple shop. By shop, I mean one of a number of street shacks in Southeast Asia.

The price is less than half the cost of Apple doing it and I'm pretty sure it'll go fine. But I thought I'd check in with you first. What can I check for to see that nothing else has been replaced in the phone? For example, the battery or camera? What else? Is there a way to quickly verify that those things haven't been replaced?
If you are implying that the service guy stole or will steal your iPhone parts, they won't. Because the products they do replace it with is actually the real deal, so your parts is not more genuine than the parts you can buy from eBay or Alibaba. The iPhone is made in China, and these China factories sell these items themselves online. I bought a few products from China and compared with the broken items and they were identical. 

Apple will obviously overcharge you for the same product and service, because it's Apple :P The guy on the street can't charge as much otherwise he won't get any customers, he needs to stay in the market. If he has a shop in a shopping mall, he can obviously charge more. But the products are the same, they just make more profit.

To answer your question, the only way to see if somethings been replaced is to see if the quality is the same. Check if the camera is just as good, battery lasts long, etc. Or open the phone and take before and after photos?

If you don't trust anyone, just buy the display from eBay or Alibaba (or iFixit) and fix it yourself with this guide on how to replace the display. It's really simple actually.. :)