As long as the exchange is a registered company, I don't mind giving my personal info since it's a important thing and can be a good help when there is fraud happening or any thing related to that that involves your account. Not an assurance though that everything will be fine but it's not wrong to consider that feature.
If you actually believe that your information's are not being used by a third party then you should think twice. You also start by simply providing your ID and you finish up by getting your account closed for some reason or having legal issues just coinbase got them lately. It's just a matter of time until bitcoin gets fully regulated and people won't like what's gonna happen If they don't control their private keys.
It is a matter of time when every government will have a regulation to have a "know-your-customer" policy in place for every exchange or platform using bitcoins. But as of now that would not be the case and there would really be little the governments can do right now to regulate the bitcoin services.