> In general, I would think this is largely up to you - i.e. only the rights you explicitly granted.
Do you have any examples of an NFT that grant rights accompanied by TOS terms for minting the NFT that affirm that it is the original artist or someone who is authorized to do so who is creating it?
> However, I believe in the US, there is no right to even download a copy of a file on the internet for purely personal reasons.
There is de minimis which isn't so much a right but rather a "this is beneath the concern of the law".
As to the US aspect, every country that is a signatory of the Berne Convention ( https://en.wikipedia.org/wiki/Berne_Convention ) recognizes the minimum standard for protection which includes the right to to make replications or display to the public that is held as part of copyright.
For an NFT, would a baseball card be an acceptable comparison for what it represents in the real world? To that end, I would contend that the same thing happens to it when the player that it represents (there is no ownership conveyed in a baseball card beyond the card itself) leaves the game.