The largest complaint by far is that the URLs for AMP links differ from the canonical URLs for the same content, making sharing difficult. The current URLs are a mess. They all begin with some form of
https://wwww.google.com/amp/before showing a URL to the AMP version of the site. There is currently no way to find the canonical link to the page without guessing what the original URL is. This usually involves removing either a .amp or ?amp=1 from the URL to get to the actual page.
Make no mistake. AMP is about lock-in for Google. AMP is meant to keep publishers tied to Google. Clicking on an AMP link feels like you never even leave the search page, and links to AMP content are displayed prominently in Google’s news carousel. This is their response to similar formats from both Facebook and Apple, both of which are designed to keep users within their respective ecosystems. However, Google’s implementation of AMP is more broad and far reaching than the Apple and Facebook equivalents. Google’s implementation of AMP is on the open web and isn’t limited to just an app like Facebook or Apple.
I’ve played with AMP, Facebook Instant Articles and Apple News. Apple News and Facebook Instant Articles work great, mostly using RSS feeds and not needing many changes to your look. But that’s not my biggest issue with AMP.
My biggest problem with AMP is not having to do special webpage layouts that include inline CSS.
My issue is canonical URLs. With things like Facebook Instant Articles and Apple News, the canonical URL for each story remains on the originating website. While with AMP, the canonical URL is on google.com.