Sally - From time to time, I've also been surprised by some of the similarities I've seen. That said, I've also seen many significant differences on searches I'm familiar with.
Bing, eg, appears to give an almost ridiculous amount of weight to a domain name, and it seems to be a poor judge of inbound link quality. Google has a much larger index, and appears to dig more deeply into page content.
It sounds like you haven't seen this thread, prompted by a SearchEngineLand story, which popped up in February 2011. On the surface, it gives some credence to your thoughts....
Google Sting Exposes "Copying" by Bing
A few days later, SearchEngineLand published a discussion of Bing's response... Bing: Why Google’s Wrong In Its Accusations [searchengineland.com]
I've seen enough differences that I'm convinced that Bing isn't borrowing. Obviously though, both engines check each other out and don't ignore obvious hints, but there's no one-to-one direct influence on individual searches.