Google's John Mueller confirmed on Twitter that technically, you do not need structured data for Google to apply rich snippets in the search results to your site. John added that although it is not needed, it would be rare to see a search result with rich snippets that does not also use structured data. So obviously it is recommended that you use structured data on your site so that you can help ensure you get rich snippets in the Google search results.
Here are the set of tweets around this conversation:
hi @JohnMu - i recall there was a video where u said basically structured data not needed for rich snippets. any chance u have the url?
— Bastian Grimm (@basgr) October 31, 2017
Yeah, though afaik rare; If you really want them, I'd just use structured data - it's cleaner.
— John ☆.o(≧▽≦)o.☆ (@JohnMu) October 31, 2017
Of course, Google doesn't want to be dependent on structured data for this but they are at where they are at now. In the future, who knows but for now, if you want to see rich snippets, I'd go with structured data markup for your pages.
Forum discussion at Twitter.