https://lockbox.relays.land
Your notes will only be visible to users that you follow, but they need to be using an outbox style client or know to look there. Replies are sent to read relays, so conversation will still move to other relays & become open as people respond and interact.
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If you use the DM style how would anyone other thank the poster person see the replies? I expected that interactions would be limited if doing this on Nostr.
Lockbox isn't a relay for group DMs, as described in the other comments here. Notes written to it would show up only in the home feed of anyone who follows you that you also follow. It doesn't accept any replies. Those are sent to your read/inbox relays, which is where it becomes more open... unless you receive replies to one very strict inbox, which isn't very common.
I think it is as close to what you initially described as anything that is currently available, though.