Not without your permission!

No data syncs in Android without your permission. Entering your Google account at first boot is completely optional. In that case information is only stored locally on the phone and you have to use adb to backup.

If you want to sync calendars and contacts over the air to your own server, there is third party software available for this. It works beautifully.

I ran my phone without the account and the functionality loss I suffered was the Play Store. It's a pain installing software manually. There is multiple replacements, including F-Droid for free software, but none that come close in number of packages offered. Other services, such as Maps, works.

So "big G 0wns your data" is really a red herring. Android is perfectly usable without a Google account, something that can not be said about Apple, where the iPhone is pretty much an expensive brick without an iTunes account.