Must be Apple, because I get missed notifications anytime I have the phone on and miss a call, using basic voicemail or Visual vm. I get VM notifications without data on basic but have to have data to receive visual vm notification of an actual voice mail having been received.
I spoke to several tech people from Verizon, they have told me that data or WiFi must be on to recieve notifications for missed calls. I had a Iphone 4 and never had an issue. Problem started for me in November with purchase of Iphone 6.
It is because WiFi goes off when your iPhone goes to sleep. If you have wireless data turned off, you will then have NO data connection, so those types of notifications will not be received.
I also remember reading somewhere this came with the latest ios update which is not compatible with iPhone 4. A few others were complaining. Most people leave data on, so do not notice. I agree you should have the option and it should work, but this is more an Apple decision than Verizon.