As a practical matter, rather than a legal one, declining to give information to your mortgage servicer seems exceptionally short-sighted.
Sure, when servicing is transferred, the old company is supposed to send over all the information the new company needs. And in the vast majority of cases, it does and everything is happy. But even if the information transfer process is 99.9% accurate end-to-end (and I'll wager that the actual accuracy rate will be lower), if the old servicer transferred servicing on 10,000 loans to the new company, there are 10 unfortunate people who have errors in their data. If you're one of those 10 people, you'd much rather catch and fix the error now rather than trying to fix it later after something has gone wrong.
If for whatever reason the new servicer doesn't have accurate information on your insurance and your mortgage agreement requires you to have home insurance (which the vast, vast majority of agreements will), then the new servicer will force place a new insurance policy. That's likely to be way more expensive than your current policy. Sure, you could contest that after the fact and you'd hopefully be able to get a refund of the fees you paid. But that's going to involve way more time and effort than simply downloading your policy document from your insurer's web site and forwarding that along to the new servicer.
As for privacy, your loan servicer is supposed to have a pretty large amount of information about you. Sure, if they're asking you to fill out a bunch of information that is obviously unrelated to servicing a mortgage, you've got a legitimate gripe. But if they're asking you to verify that there wasn't an error in the name, address, SSN, etc. they received from the previous servicer, you're not giving them anything that they shouldn't already have. And as a practical matter from the new servicer's standpoint, this is a good time to have people re-confirm contact information because lots of people will have new email addresses, phone numbers, etc. that they didn't bother to inform the previous servicer of.