Actually Ampie used to work differently: Ampie would store the local cache of links from your twitter feed and links submitted to HN, and check against it when you went to any web page. If there were hits, it would show a small tab in the bottom right corner, showing that there are N tweets, M HN submissions with this page. Then you could click on the tab and the sidebar would pop up, so no requests were sent until you actually clicked on the tab.
There are two problems with this: first, the cache had to be large, several hundreds of megabytes, because there are a lot of links (~200k from HN, and each person I follow on average tweeted 1k links among the last 3k tweets that twitter lets me access) and indexeddb is not memory-efficient. Second, this is easy to miss, and those numbers alone don't give enough information to go through the effort of clicking there every time. And this doesn't include the backlinks and HN comments mentions (fun fact: there are about 4m URLs mentioned in HN comments), so you'd miss out on a lot of stuff. You could suggest Bloom filters as caches, but that is a bad solution since now you only can show if there is a hit or not (instead of two numbers), but also there are false positives so the sidebar is not reliable anymore, and given that you go to hundreds of URLs a day, the FP rate has to be really low which makes the cache really large. Maybe using something like https://jlongster.com/future-sql-web would have reduced the memory usage, but at that point I gave up on that approach already.
Given how technically hard it was to maintain those caches of links (they had to be updated incrementally, for example, not to send hundreds of MB every time they were recomputed, you had to wait to download the cache after installing ampie), the fact that most of the friends I have talked to didn't mind if the request was sent on every URL visit (as long as I promised not to save it), and the fact that this made the sidebar less fun, I decided to remove those caches and just send the URL to the server on every visit.
So yes, every URL you go to is sent to the ampie server, but I do not save the URLs sent to the server, and if ampie is disabled on a domain (you can disable it on a domain in the settings), it will not request information about pages on that domain at all.