A lot of us use Discord for chatting, so I thought I'd address some privacy basics.
There's no such thing as a truly private conversation on a Discord server. The server administrator, and possibly other individuals of the administrator's choosing, can see EVERYTHING posted on the server. Even if a limited-access channel is created, they will still have the ability to see. To see who has access to see a conversation in a channel on a Discord server, click the "Member List" (looks like two heads/torsos) at the top of the screen.
However, sending a Direct Message (DM) to someone is totally private and cannot be seen by the server administrator or anyone else besides the intended recipient (unless someone else has access to their account). To send a DM, just click on the name of the person you'd like to send it to.
If you add someone as a friend, you can continue to DM them even if you are not on a server together. See https://support.discordapp.com/hc/en-us/articles/218344397-How-do-I-add-my-friend-to-my-friends-list- for more information.
If you want to have a truly private group conversation, you can do a Group DM. You must be friends with someone in order to invite them to a Group DM. See https://support.discordapp.com/hc/en-us/articles/223657667-Group-Chat-and-Calls for more information.
The only privacy downside to friending someone on Discord is that they'll be able to see if you're active, so don't hesitate to friend people you'd like to stay in contact with!