You currently work for company A, correct? Then it is very unethical to reveal to company B information given to you by company A as part of its day-to-day business, including (you might say especially) regarding hiring or other HR matters. The fact that you have a job offer in hand from company B is completely irrelevant, it would be a failure of your duty to company A. This is the kind of behaviour that leads to companies escorting employees directly off the premises when they hand in their notice -- they've been burned by or heard stories of people doing things while working out their notice that they would never otherwise contemplate.
It may be that company B expects you to reveal to it any information you have about its employees that it would find interesting: especially once you have started work at company B. In particular, going for interviews might be seen as important information since it suggests that employee is planning to leave, and this is something the employer wants to take into account.
Personally I think it would be pretty shady to extend this to the level of expecting you to share information from your old employer. Hiring someone doesn't give you the right to all their inside information. Nevertheless, if company B is pretty shady (in my view) and you want to work for them anyway, then you'd have to take their expectations into account. If company B is not shady, and the thing you're worried about happens, then you say, "I received confidential information at company A and of course I could not pass it on to you". If you say that and company B holds it against you, well, you have a problem with your employer.
I think a closer call (and one that I don't know the answer to), is this: assuming the member of your new team doesn't get the job at company A and is still there when you start work, do you mention to them that you know they applied for the job, in order to ask them whether they're unsatisfied at company B and what would change that? In favour of it, you're honest and open in what you know about your team member. Against it, it might be seen by the employee as applying more pressure or suspicion than you want, and you might have to put on record for company B what you've discussed.