I recently interviewed at both Google and Facebook. At Facebook, the hiring manager was really curious since I was recruited for a similar role and wanted to hear about my interview at Google. I declined to respond but was curious what would be appropriate to say in this situation next time. He was persistent with his questions and it was borderline awkward for me to keep resisting.
This reminds me of a company that recruited me because I worked for a competitor that became the envy of the industry. As soon as they had me lassoed in, they literally asked me these really personal questions, like "what were XX's financials on a monthly and quarterly basis? What did they make last year? How did they do it? What were the strategies? What technologies were in play? " It crossed so many lines. I happily walked away from them after that. (This is a completely true story, btw.)
Here's the difference between my situation and yours: you are not obligated to disclose anything, but there's no reason you should withhold anything either. Unless you signed a confidentiality agreement, you're not legally bound to keep quiet.
There are three ways to navigate a situation like this:1) Tell them what they want to know.2) Tell them what they want to know, but keep it very non-specific.3) Withhold from saying anything.
I would probably stick with #2, since it would show a willingness to cooperate. "I don't want to give specific details, but the high level was that they wanted...." It doesn't leave a bad feeling and shows that you want to be discrete. It communicates that 1) you're a team player and 2) you are able to be trusted. The other two options show only one of those components.