After a video is finalized and processed, you will see a URL on your Big Day page. Sometimes people just share with the gift recipient but usually, it's shared with all Contributors. Copy that URL and send it to whomever you choose. Anyone clicking the link will be able to view the video.
Please note that anyone with the link will also be able to forward it to whomever they choose. So if you prefer to keep it 100% private and locked down, just download the video to your computer or mobile device and share it directly with your lucky gift recipient.