Here's a slight deviation from my standard "so you're an RPG writer" advice, but it touches on the two pillars of my professional life: gaming and the law.
There's been a big push in actual-play podcasting in the last few years. I'm even part of one myself (it's an awesome Starfinder campaign, and you can
find our episodes here). With more people meeting virtually, I've heard more people considering recording and uploading their games as actual-play podcasts. This is fun, and the tech available makes it easier to do than ever. I note that such podcasts are rarely "actual play," but are often edited a little (or a lot) to make them seem faster paced, more lively, added sound effects, and so on.
A lot of this advice goes into streaming games as well; these are rarely edited on the fly, of course, but they implicate many of the same legal issues.
If you're thinking of making your game an actual play podcast or a stream, you should of course involve your players and let them know of your plans. With all the actual play podcasts and streams out there, you might not get a huge listenership, but you might develop a dedicated audience over time. You'll probably want an agreement among yourselves about rights and expectations, to address any hard feelings if a particular player isn't happy with a session (or a bunch of sessions), or wants compensation retroactively, or whatever.
Here are the elements of the kind of agreement you might consider. Generally, this type of agreement is called a "Media Release Agreement" or "Release Agreement."
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The Parties. Invariably, one person--usually, but not always, the GM--is going to be spearheading the drive to podcast the group's sessions. This person is going to undertake the effort of recording, editing, posting, and all that (the specifics of which are, frankly, beyond me, and you can probably find great advice about all that elsewhere). This advice is really for you, if you're that person. You should consider the agreement between you--as the person doing the legwork to make this all go--and the other players, who are participants in the podcast (we'll call them the "Participants" in fancy capitalized legalese). You might not go through the effort of establishing limited liability company or corporation for your podcast, but if you do, the agreement would be between that entity and all the Participants. It's fine to have all the Participants sign one agreement collectively.
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Participant Authorization.
This is the first of two core reasons for this type of agreement. Each Participant should grant you permission to use their appearance (including any video, images, sound, recordings, and so forth) however you deem necessary, and edited as you see fit. You should also state that there's no specific obligation on you to use any particular Participant, so if you edit someone down pretty significantly they don't have grounds to complain. This seems heavy-handed, but it allows you to mix and edit to keep the podcast lively. If you have someone who insists on being the center of attention and wants to be in the podcast so they can be the focus all the time, at everyone else's expense, then maybe making a podcast about the whole gaming group isn't the right thing for you to be doing in the first place.
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Participant Release.
This is the second of two core reasons for this type of agreement. Each Participant should release you from any claims relating to the appearance. Someone might later claim there was defamation, an invasion of privacy, infringement of rights of publicity, or the like. You won't automatically deflect every such claim should one of the Participants decide to make a big issue out of this, but you want to have this release at the outset. That said, these gamers are your friends, so if someone says, "hey, can you edit out that part during the ogre fight where I yelled out in surprise and then flushed the toilet?" you should probably just do that before they're known across social media as "Grobnar Lightfingers the Flushing Rogue."
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No Right of Review. Some jurisdictions might give a Participant the right to review or inspect anything that goes out with their likeness; you should state that there's no such right for the podcast. Otherwise, you will really slow down your release by waiting for everyone to weigh in on each recording in its entirety.
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Consideration. It's a fundamental tenet of contract law that each party must receive something from the other for a contract to be valid; a purely one-sided contract can be deemed unenforceable. For release agreements like this, it's very common--and, arguably, necessary--to include a statement about what the Participants get out of it. Even a statement that the Participants are being allowed to participate or perform in the podcast, and that such participation is agreed by the parties to be consideration, is a really good idea. You should state that there isn't any other compensation at all for their participation. (If there is, then you're looking for a slightly different type of agreement about hiring talent for performance, which has separate legal issues.)
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Boilerplate. Standard contractual terms about amendment, waiver, notices, governing law, and so on should be part of this contract, just as it is with any formal legal agreement. I could (and have!) blog about all these provisions that everyone besides me finds super boring.
With the legal technicalities out of the way, you can get to the practical technicalities--which are much more often the biggest hurdles to overcome to get an actual-play podcast going.