Is there any official guidance on where to put a transcript for a video? Should it go on the same page as the video or on a separate page?
Transcripts and videos are one in the same thing. Having them on a separate page can be quite a frustrating experience.
Let's take a look at other examples
Lynda.com has the transcripts right below the video:
This allows for easy reading and watching at the same time. And the current line is highlighted for easy access.
Youtube had the "transcripts" right in the video, so no need to scroll if you have a smaller screen.
I recommend easy access so you don't have to have the user jumping from place to place.
Putting it on the same page makes a lot of sense from the navigation perspective, as well as from the SEO angle.
If you put them on separate pages, the page with the video will never show up in the search engine results, since it has almost no text content. So, from the search results, people would get directly to the transcript. And then they will need to figure out how to get to the video itself - extra step that could be avoided.
While there are tools that auto-follow the transcript while the video plays, I am not sure whether their text content can be indexed by search engines - this is a big question. So, as a simplest first step, I would just put the transcript below the video.
An additional thing for easier text<->video navigation is putting time stamps to the main parts of the transcirpt. This will help readers jump to the needed part of the video if they are interested in that part only.