Yesterday Twitter launched a new video sharing app called Vine. The service lets you compose and share six-second videos using its iPhone app., currently there's no way to set any of your videos, or even your profile, to "private." Any videos you upload are going to be public; viewable by anyone using the service.
There is a way to create a video using the Vine app and share it privately, however. It's not an ideal workaround (that would be privacy settings in the app) but it's an easy way to still be able to get creative with Vine and share your work with friends without the rest of the Vine community seeing it.
What you'll need to do is create a Vine as you normally would. Once you tap "Next" to go to the share screen, the video is saved to your Camera Roll. Since the video is already saved, you can either turn off the "Share on Vine" switch and tap "Done," or go back to the compose screen, tap the X, and delete the post.
With the video saved to your Camera Roll you can then share it -- privately -- through MMS, iMessage, e-mail, and so forth.
Again, it's not an ideal sharing method, but it does allow you to maintain your privacy should that be a concern.