There is no perfect solution for that in an automated way. Think e.g. of an app intended to sync your contacts or calendar data with a cloud service: it must access both, your data and the Internet, and it must transfer your data (that's it's purpose) – so how should a "monitor app" tell the good from the bad? And that's only one combination, thus not even a "simple whitelist" would do.
However, there are possibilities. You could use a Stand-Alone Permission Manager (requires root), revoking corresponding permissions from the app-in-question (e.g. access to those private data), or using an Internet Firewall app (again, the good ones require root) to forbid that app accessing the internet. Both ways may result in restricted functionality. So these solutions require some knowledge of what to restrict where (though try-and-err might work as well: if something breaks, just lift the restrictions a bit). Several permission managers also keep logs of which permissions an app tried, so you could check then what might have caused functionality to break.
One such app (and a very good one) is X Privacy, requiring root and being based on the famous Xposed framework:
XPrivacy (source: Google Play; click images for larger variants)
As the name suggests, this app is intended to protect your privacy. It monitors which permissions an app accesses, and allows a very granular control. Experts can even go beyond the permissions, and e.g. restrict parts of it while permitting others – while beginners can stick to a "simple mode" (the "full potential" otherwise can become quite overwhelming).
Forgot to mention: Xprivacy (like some of its competitors) also can "ask you on access", so you can a) decide context-based and b) learn at what moments such access happens. This way you can figure whether an app accesses your private data without your consent. I e.g. remember an app that wanted to access my contacts without me triggering an action requiring that – which is exactly what you want to figure out for your apps, and thus a "perfect match".
Hopefully, Android 6 (Marshmallow) brings this feature out-of-the-box – as it was announced to ship with a permission system allowing adjustments after install. I'm just sceptical on "how deep that reaches" – and especially doubt it will e.g. allow you forbidding Internet access to apps (lost revenue in ads certainly will prevent that from Google's end).