There are two main concerns: copyright restrictions on distributing stuff that other people made, and publicity rights to use other people's picture and voice to promote a product. Both restrictions reduce to the fact that you need the rights-holder's permission to use the material. It is not required that the permission be written on paper or that it be signed, though those are two of the most bullet-proof ways of establishing permission. If you claim that you have permission, the court would want to see that permission, which would be some statement by the rights-holder that they permit you to do the thing that you intend to do. It is not enough that the person has emailed you a video, because they might not know that you have in mind showing it on television. It is also not quite enough that the email they send you has some text somewhere (which you write), saying that you intend to use the video in such-and-such way. The question would be, did they actually see and understand that bit of text, or might they have misunderstood what your intent was? On a web page where you click "I agree", the agree button needs to be next to what they agree to so that they don't have to go hunting for the terms. Same would go with an email.
I suspect that an email (from you) like this would suffice:
Can you send me a 1 minute testimonial video about our product, which
we will use for advertising? We will use your real mane, we may modify the video, we will use it
on television, and post in on the interwebs. At any point if you want
us to stop using the video, email me at email@example.com or
call me at 234-567-8910. Please reply, including this email, and say
"Yes, I agree", above your signature. Also please include your real
You need to get a lawyer to draft this if you want to go this way: I just want to point out that you're making a big and wrong assumption about what they have agreed to, by sending you a video. You need to say explictly what you're asking permission to do, and then you shouldn't do anything that you haven't been given permission to do. You need explicit permission: people do things for all sorts of reasons, so while it's reasonable to assume that a person wouldn't give you a video in response to a solicitation for a testimonial, you should think in terms of what misunderstandings could arise.