SayClip is a video messaging tool that uses the contact system on your iPhone to send fun, short videos to specific people. While there are issues in the setup phase and some performance issues when it comes time to render edited videos and send them, the app generally works quite well.
Setup takes some time and requires you to integrate with Facebook, which is admittedly an unnecessary requirement for an app that then relies on your phone's contact list. But despite this Facebook reliance, the app rarely accesses or interacts with that data, staying largely phone-bound. You will have an inbox, a sent box, and then a camera app to create new messages to send. You can create messages from scratch or you can take an existing video and add voice over to it. This is particularly useful when describing an event or a vacation spot you recently visited. Of course, your friends need to have SayClip installed to receive your videos--otherwise they simply get an invite to download the app.
Because of the restrictions placed on video sharing over more SMS networks, SayClip is a useful tool that allows you to have one-on-one conversations through video on your iPhone. While there are certainly some small issues here, the app as a whole works quite well and will provide the platform you've been looking for to replace blurry media messages or semi-public Vine or YouTube videos.