Does iTunes WiFi sync communicate with the Internet? When I use WiFi sync using my wireless router, is the data transfer completely in local network OR does it happen over the internet to iCloud or another service? Will the bandwidth used be deducted by my service provider?
If you iPhone talks to the internet while connected to WiFi, then yes, it costs your home internet bandwidth.  If it talks to the internet outside of of WiFi range, it costs your mobile internet bandwidth.

Syncing your phone with iTunes does not use the internet, it only uses the local network.  If your phone is configured to sync with iCloud, then it is using the internet.