In order to create a video for someone else, you have to go through some steps. Go to the Video Messages screen and select the video you want, customize it and then click "create this video".
After the video has been created go to My Video Messages and select the customized video you wish to share.
When selected you can click to play the video as usual, or you can click on "Share with friends or family".
A screen will appear where you can choose through which application you want to share the video. Recipients will receive a link to a web page where they will be able to view the video.