This is true, and they do. I'm replacing Comodo's Valkyrie API with Symantec's Browser Protection API for performance reasons. But, no data is logged, and only the minimum amount of data is sent to each provider.
Osprey is entirely asynchronous. It doesn't hold pages at all. If you visit an unsafe website, you might see it for 50-250ms before the connection is stopped.
1) Osprey is just more protections built into one. It combines seven pre-existing safe browsing extensions you can already download into one lighter and better extension. All these extensions do is send the URL you visit to an API. That's what SmartScreen and Google SafeBrowsing do.
2) Read gorhill's comment. Checking network activity is easy, and checking every extension is always good.
Thanks gorhill. Yes, this is true! Osprey only sends its stripped-down URLs to the services you turn on. Nothing more. It's as anonymous as it can be without using a VPN, afaik.