Just in case you thought Google and Bing share the same index of the web - you'd be wrong. John Mueller stated the obvious on Twitter this morning. John said "we don't share indexing with other search engines." He said that when someone asked why Bing has their pages ranking in their index but Google does not.
Here is the tweet:
We don't share indexing with other search engines. I'd ask in https://t.co/NPZwEmBjV7 with your specific case.
— John ☆.o(≧▽≦)o.☆ (@JohnMu) June 26, 2017
There were some bugs in the past where Google would crawl Bing's cache and Bing copied Google's results at one point also. So there is some truth to the question but no, they do not actively share an index.
Forum discussion at Twitter.