Will iphone Skype to Skype show on phone bills?

Alright, I know there's already a lot of answers on here pertaining to whether or not Skype to Skype calls will show up on a phone bill but I see a lot of conflicting information so I'm really hoping to get a clear answer--if a rep from a cell phone carrier could chime in I would greatly appreciate the insight.
So my questions are:
Will usage of the Skype for iPhone free app show on a phone bill? We both have the app installed on our iPhone 5s. Will voice calls, video calls, or instant messaging show up on a phone bill or give any information such as phone number, email or skype name? Would instant messages be able to be viewed on a bill if requested? If we're both using Wi-Fi, will that affect it? If we're both using our 4G LTE, will that affect it? I understand it would use data when not on Wi-Fi, but would this usage be shown as "Skype" on a bill, or just lumped in with everything and show no differently than regular internet usage?
If a more detailed bill were requested would that change any answers? One phone is AT&T, the other verizon.
I'm needing an app that will not register voice or video calls or Instant messages between two skype names (not numbers) on a bill. I want to convey the need for discretion is in no motivated by anything of an explicit nature--it is just a serious situation that requires communication between involved parties leave no trace.
I appreciate any useful information.

No it won't

There's no possible way for it to show up on the phone bill. Your carrier has absolutely no access to Microsoft's servers.