That's not entirely correct – examples are really only stored in the database when they're annotated. Here's how the stream works: when you open the Prodigy app in a browser, the app will request the next batch of examples from the stream. This is just a slice of the generator. When you open the app again in another tab, it will ask for the next batch from the stream. The same happens if you refresh the browser or close and reopen. Until the server is stopped and the session ends, Prodigy won't know whether the examples that were sent out are coming back – maybe session 1 takes longer to annotate, maybe their connection died. When you restart the server, Prodigy will know which answers it has, and which examples in the stream haven't been annotated yet, and will send out the unannotated examples again.
If you set force_stream_order, Prodigy will keep a copy of the stream and re-send examples in the exact order they came in, and re-send batches until they're answered. So when you open the same session in a different tab or refresh the browser, the same batch is sent again. Also see this thread for some background: Missed examples on prodigy interface - #2 by justindujardin
Yes, you'd just have to register the database as shown in my example above, then it should work.