Google Drive is full of options that often go overlooked. One of those options is privately sharing videos. To share videos through Google Drive upload them to your Google Drive account, preview them, then share by using the sharing options at the top of the preview screen. The sharing options allow you specify who can access the video. The screenshots below outline the process. (Click the images to view them in full size).
Step 1: Upload your video to Google Drive.
Step 2: Select you uploaded video.
Step 3: Click the share button.
Step 4: Choose your sharing options.