DuckDuckGo really only protects your privacy for the moments that you're on their search page (basically when you see their logo in the top-left corner and see "duckduckgo.com" as the domain name). This is similar to how Google most directly violates your privacy when you're on their search page, branded with the Google logo "google.tld/search/*." (Replace ".tld" either with ".com" or your country's domain ending).
Google, however, reaches farther. They do unbranded powering of site specific searches (I don't know what CNN's search uses, but for all I know, it's Google). When you change the default search engine in your browser, you're simply changing what URL or domain you direct your search to (in Google's case to "https://google.com/search/?q=[query]
"; in DDG's case "https://duckduckgo.com/?q=[query]
"). You could personally go to Google and search from their search page and you would, in that case, get Google results. If Google were your default, you could go do DuckDuckGo or Chrome and get theirs. And you can go to CNN with any default you choose and still get results from whatever search engine CNN uses.
I love reading ToS (seriously, I do) and if you're concerned about your privacy on a website, you should too. However, if you want to generally protect your privacy, I would suggest:
- If you haven't already, get a browser like Firefox or the TOR browser.
- Get Ghostery and/or PrivacyBadger; These add-ons block third-party trackers, meaning that when you visit CNN.com (which has at least three trackers) it blocks and lists all three of them. However, tracking done directly by CNN.com is not blocked.
If you're feeling really good, get TOR. (I haven't yet done this step, but I plan to some day). The NSA has called TOR "the king of internet privacy" and TOR certainly deserves it. It sends all your requests through a complicated series of encryption layers, making sure that the end website has no idea where the request came from.
If you need any help or want to ask any more questions, please do.