There is no unambiguous prohibition in the Geneva Conventions about releasing videos of POWs. It's difficult to prove a negative, but I think we can defer to common sense here; to begin with, North Vietnam then and Ukraine today would not have released those videos if doing so was an obvious war crime. But also, at the times the conventions were drafted (the last in 1949), video interviews were unheard of or incredibly rare.
But there is a concern broadly that under the right conditions, the release of videos of POWs can sometimes violate Article 13, which prohibits subjecting POWs to "insults and public curiosity."
Art 13. Prisoners of war must at all times be humanely treated. Any
unlawful act or omission by the Detaining Power causing death or
seriously endangering the health of a prisoner of war in its custody
is prohibited, and will be regarded as a serious breach of the present
Convention. In particular, no prisoner of war may be subjected to
physical mutilation or to medical or scientific experiments of any
kind which are not justified by the medical, dental or hospital
treatment of the prisoner concerned and carried out in his interest.
Likewise, prisoners of war must at all times be protected,
particularly against acts of violence or intimidation and against
insults and public curiosity.
At the time this was drafted, this would have been understood to mean literally parading POWs on public display for propaganda reasons. But in the age of mass media, some have raised concerns that POW videos serve the same purpose. Notably, in 2003, the Human Rights Watch criticized both the Iraqis and the Americans on this basis:
This provision protecting POWs from “public curiosity” appears to have
been violated by both the Iraqi and the U.S. governments. The Iraqi
government has filmed American POWs and interrogated them before
cameras. The U.S. government has taken insufficient measures to
prevent journalists embedded with U.S. forces from filming Iraqi POWs
held by the United States.
U.S. Secretary of Defense Donald Rumsfeld has appropriately criticized
the Iraqi filming of American POWs. However, he has said nothing to
date about the filming of Iraqi POWs by media operating alongside U.S.
forces.
There's no clear red line of when a video becomes prohibited, but the surrounding language suggests the provision is about protecting prisoners from humiliation or intimidation. If a POW is genuinely okay with giving an interview or being on camera, and if it's not being done in a way that would humiliate him, it's probably above board. The reasons some have concerns about the Ukraine videos is it's pretty much impossible to determine in real-time when these sorts of videos are coerced.
To some of your other questions:
Does it matter if it's state TV or independent media? I would think no, because the provision in question demands that state affirmatively protect POWs, not just from themselves, but third-parties.
I've seen people saying that you can't violate the Geneva Conventions if you're not a nation since private citizens and
corporations, but if that's true then why do so many non-nations have
to worry about accidentally using Red Cross symbolism? Because the
countries themselves have an obligation to stop private parties from
improperly using the Red Cross. From the 1929 conventions: "The Governments
of the High Contracting Parties... shall adopt or propose to their
legislatures the measures necessary to prevent at all times:(a) The
use of the emblem or designation “Red Cross” or “Geneva Cross” by
private individuals or associations, firms or companies..." So if I
use the Red Cross improperly, I get in trouble with my nation. If
my nation doesn't take steps to prevent it, they're the ones
actually violating the Conventions. But I can't actually violate the
Conventions.
If taking the footage itself is a violation, what about sharing it or rebroadcasting it? You personally are not a signatory to the
Geneva Conventions, so you're fine on that count. I suppose it's possible it could violate the letter of the laws of some countries. Even so, I'm skeptical any liberal democracy would take action, as it could discourage reporting on the conditions of POWs, which is a net good.
