No it does not - Only data transfer on the public network interface will count against your bandwidth allowance.

From the documentation:

Any data transfer sent by a Droplet using a public network interface will count against the transfer pool for that Droplet’s account. All IPv6 traffic uses the public interface. Data transfer between Droplets over the VPC network uses a private network interface.

https://www.digitalocean.com/docs/billing/bandwidth/

Also - you can use this bandwidth calculator to estimate your bandwidth allowance and overage cost based on Droplet configuration.