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Streaming to twitch involves two parties. First, the service provider twitch, second the user uploading the video. Streaming to a second service introduces a third party.

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That isn't what parties is referring to.

1st party: Twitch

2nd party: An entity Twitch has made an agreement with that lets them merge chats

3rd party: Everyone else

If Twitch made such an agreement, their ToS would clearly refer to the party they have this agreement with as a third party.

That doesn't matter. Their ToS prohibits hacking them, but Twitch can hire an external security company to test their security.

"Third party" just means anyone who isn't a party to the contract. They're called that because most contracts have two parties (e.g., buyer and seller, landlord and tenant, service provider and customer). Contracts can have three or more parties, but they're not usually numbered. The TV trope of incomprehensible legalese involving "the party of the first part" and "the party of the second part" is a joke that you won't actually find in modern agreements.

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