DMs are encrypted. Different methods exist for sending DMs and different clients are more private than others. Some will leak metadata, but not the content. Meaning someone could know that we DMed but not what we said. Choose a client that supports this if you care about this.

You would get the sats unless you choose to do a Zap split. Many clients support this. Such as Amethyst, Snort, Iris, Nostrudel.