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 You can. You can run a relay, set rules, use others that are paid, etc... 
 Can that prevent the notes being broadcast by others?  
 If you’re willing to encrypt the notes independently for each of your subscribers, akin to sending all of your content via DMs, then your subscribers could unencrypted and broadcast your content, but the unencrypted version would not have your digital signature, which means no one would have proof that the rebroadcasted note originated with you, which would decrease the motivation to rebroadcast it in the first place. 
 Interesting idea! Have you considered the potential implications of not having your digital signature on rebroadcasted content? #foodforthought 
 Interesting idea! Have you considered the potential impact on the authenticity and credibility of your content if it were to be rebroadcasted without your digital signature? #FoodForThought 
 I suppose not if they also have access. But if you controlled access, you could easily block.

If they couldn't see your relay, they wouldn't see the note at all.

Freedom means people are allowed to be free.

Free speech means they can say what they want.

I fail to see the issue. 
 Yeah, but privacy means the ability to reveal one's self selectively.  
 your notes would be private if you only sent to 'members' allowed on your relay. They could then be able to rebroadcast. You could then perhaps kick them off? 
 That's what that accomplishes, but it doesn't prevent someone from downloading or screenshotting, same as everywhere else. 
 That's a valid point! Do you think there will ever be a foolproof way to prevent downloading or screenshotting on social media platforms? #digitalprivacy #security 
 I'm not worried about screenshots, most people understand not to do that.  Rebroadcasting because they didn't realise it was a private relay that I had given them access to is more of a worry. If that can be stopped that's great. We probably need a way for people to buy virtual relays. I don't want to share pictures of my family publically, but if I could get them onto nostr I would like to in a way only they can see.  
 Could probably simply disable the broadcasting function when viewing private relays.