There are 2 types of direct messages. One is a spec that is (hopefully) being depreciated. Primal uses that one. Amethyst supports both the dying bad one and the better one. If you go into Amethyst, click my profile to send a mock dm, or send one, i don't mind... you'll see the little incognito man. You can switch back and forth between the message types. Active means they're more secure. Amethyst will automatically set the msg type, based on what type your recipient is using. If you do msg me, then go look on the npub sign in, you won't see the msg data.

As far as everything else, nostr was designed to be censorship resistant above all else, so most things are public by default. You can utilize nostr in less public ways, once you get familiarized with choosing, setting up, and navigating relays. As with anything on the internet, there is no guarantee of privacy. The difference with nostr is that there is no 3rd party to mitigate these matters on your behalf. You are responsible for your freedom, your privacy, your content, what you view (and not), and what you choose to share.