First, go to "My Live Events on YouTube. Creating a new live event, or, if you want to record now, simply "Go Live Now".
Fill in the information for your event, and press "Create Event". Make sure to choose "Unlisted" if you don't want it publicly available. It will still be accessible to anyone with the link.
At the schedule time, go back to "My Live Events", press "Start Hangout on Air"
A Google Hangouts window will pop up. From there, you can share your screen, and do anything else that you would from within a Google Hangout.
After you've started, you can send the link to the video to whomever needs to watch. You can get this link from "My Live Events" by right-clicking on the video link.
After the video is done, you cand easily download the video from YouTube, to upload elsewhere.
Simply go to "My Videos", find the video in your list, select the dropdown next to Edit, and click "Download MP4"
You can also use YouTube's Video Editor, and make changes afterwards, and then download the edited video.
If you want, you can then delete the video from YouTube, if you'd rather they not keep it.
And there you have it. Easy to stream videos, which you can download afterwards for your own use elsewhere.