How do I make my videos private on Vine?

All videos published on Vine are public, and there is no setting which allows you to make them private. Twitter has promised to add more ways to control the visibility of your content in the future.

Update: Version 1.3 of the Vine app has Protected Posts support. You can control who can see your videos by navigating to Home -> Profile -> Settings -> Your Content. Tap the "Posts are protected" slider to turn on the privacy feature. Now only people who you approve to follow your profile can see your videos. However, anyone will be able to access any videos you share on Facebook or Twitter via the web.