That's called a group chat

It's just an entirely different spec than what NIP01 is built for. It could be done but it would be extremely useless at any scale beyond a fully trusted group chat. And for what reason? You don't need a profile if everyone who sees it already knows you personally.

Privacy is a weakest link system. If a single person who has access broadcasts it, you're no longer private. This is somewhat alleviated in centralized platforms where, since the delivery layer is a monopoly, the leaker needs a separate platform to broadcast on, that normal people won't check, since their clients only read from the centralized server. With nostr, anyone you give access will be able to broadcast to any relay they want, and other users will be able to read your notes from those relays by default from any client. Nostr is unable to hide fundamental realities, much worse than a centralized platform can. The fundamental reality is, once data leaves you and goes to someone, you are at their mercy in regards to them sharing it with literally everyone. This is obviously unsustainable beyond a few close friends that you really trust, at which point you should be just using a private group chat. Centralized platforms can put some barriers in place through client side Inconveniences such as snapchat blocking screenshots, but it's nothing but a minor hurdle, and with the open client marketplace, it's completely impossible on nostr, since clients compete on providing the best experience for you. If there was a way to mark a post to be unscreenshottable, no client would implement it, and those that do would be beat out by those that don't, because people will choose the client that doesn't limit them.