Sure, noone has the expectation of privacy in a public internet game. I dislike videoing people in public without their consent but it's a game of chess.
Are you allowed to...

...record your own games with a screenshot recorder and share the game on Facebook or YouTube? I would like to do this but only if it is allowed and I want to respect the privacy of other people, of course. If anyone has any input on this I would love to hear it.
Only if you ask them and they agree to it.
To be honest, I wouldn't want to have my name visible on a YouTube channel where someone beat me without knowing it.

...record your own games with a screenshot recorder and share the game on Facebook or YouTube? I would like to do this but only if it is allowed and I want to respect the privacy of other people, of course. If anyone has any input on this I would love to hear it.
Only if you ask them and they agree to it.
To be honest, I wouldn't want to have my name visible on a YouTube channel where someone beat me without knowing it.
Tough Luck - A person has every right in the world to publish their games of chess against you, with your name listed.
When I play in a tournament, if I lose, I can't tell someone not to publish my game. Freedom of the Press.
What they can't do is take their phone and make a video of you and publish it on UTube without your permission.

Well thanks for the input guys! I found out a way to compromise the situation... the recorder allows you to record segments of the screen rather than the entire screen so I will just record ONLY the board and leave out any names and everything. I see nothing wrong with doing this now!

What they can't do is take their phone and make a video of you and publish it on UTube without your permission.
wrong
(actually, it depends)

...record your own games with a screenshot recorder and share the game on Facebook or YouTube? I would like to do this but only if it is allowed and I want to respect the privacy of other people, of course. If anyone has any input on this I would love to hear it.
Only if you ask them and they agree to it.
To be honest, I wouldn't want to have my name visible on a YouTube channel where someone beat me without knowing it.
Tough Luck - A person has every right in the world to publish their games of chess against you, with your name listed.
When I play in a tournament, if I lose, I can't tell someone not to publish my game. Freedom of the Press.
What they can't do is take their phone and make a video of you and publish it on UTube without your permission.
Because you agree to that when you sign up for the tournament, mate.
I've seen no such thing here.

Well thanks for the input guys! I found out a way to compromise the situation... the recorder allows you to record segments of the screen rather than the entire screen so I will just record ONLY the board and leave out any names and everything. I see nothing wrong with doing this now!
You can blank out their names too in the video too and post on Youtube and Facebook

Only if you ask them and they agree to it.
The moves of a game are not considered copyrighted and so the game could be shared in any manner.
Your chess.com games and screen name are already public (you don't need a chess.com account to view them). I don't think any additional consent or permission would be required. I don't think it's an etiquette thing either... lots of people stream live chess.com games as well.
I don't think any private messages should be shared but the laws about privacy of those conversations vary from jurisdiction to jurisdiction. In some cases only one person in the conversation needs to consent to disclosing it.

Only if you ask them and they agree to it.
The moves of a game are not considered copyrighted and so the game could be shared in any manner.
Your chess.com games and screen name are already public (you don't need a chess.com account to view them). I don't think any additional consent or permission would be required. I don't think it's an etiquette thing either... lots of people stream live chess.com games as well.
I don't think any private messages should be shared but the laws about privacy of those conversations vary from jurisdiction to jurisdiction. In some cases only one person in the conversation needs to consent to disclosing it.
Can always edit out the right side of the screen and leave only the board if worried about privacy laws.
I think the names like you mentioned here is ok to put online

Some guy trash talked me in a game, can I post his messages to me in YouTube?
I'm not a lawyer...
Typically you can share private messages sent to you unless there is a confidentiality agreement between you and the other person. Disclaimers sent with the message are not typically enforceable because you don't have to consent to receiving them. This doesn't mean you can post private emails from your ex to social media.
If your YouTube channel is monetized... there could be copyright issues with posting content created by someone else (but this likely won't be an issue as trash talk is likely not an original work).
I'd share it on YouTube with the screen name intact, lol. If you're worried just mask out email address or screen names.
...record your own games with a screenshot recorder and share the game on Facebook or YouTube? I would like to do this but only if it is allowed and I want to respect the privacy of other people, of course. If anyone has any input on this I would love to hear it.