What is meant by "sensitive information" here? Browsers can't just willy-nilly access your local files or something like that. The one thing I can think of is using JavaScript to collect information that can be used to identify you. (Is that "sensitive"? I'd put that in "identifying information".) My honest suggestion is to keep using NoScript and just allow as few domains as possible. The next best option is to stop using websites that break without JavaScript when there's no reason why they'd need it.
I can imagine there being a plugin that spoofs some common ways that allow sites to identify you cross-sessions / browser / websites without your consent, but blocking JavaScript (by default) is likely one of the best ways to reduce the amount of information collected about you. When you do find such a plugin, check out one of the "browser fingerprint" testing sites to see how unique your fingerprint is.
(That is, if I even understood the request properly in regards to the "sensitive information" bit.)
Can you link to a source that confirms this information can be collected with JavaScript (with browser comparison, ideally)? That seems outrageous if it was actually possible.