With Lobby Chat, only other people who have Lobby Chat enabled can see what you type/hear you. If, say, you are on a quest, and both Maxi and I are waiting in the Lobby. I have Lobby Chat on, Maxi doesn't. I can see/hear anything you type/say, and Maxi cannot. However, you would still be able to see anything Maxi might type, even though he couldn't see what you type. Though you wouldn't be able to speak to him, or hear him.
I haven't tested this as I don't have a keyboard, but apparently you can use TAB to whisper people in the lobby. If you don't have a keyboard, you can use ZR on the classic controller.
That works with a keyboard, but I didn't know which button did it without a keyboard. You use it to chat with a specific player, where only they can see what you type.
Speaking of, here's the colour-code for the chat:
White: everyone can see it
Yellow: private chat message
Green: someone in the lobby
Blue: automatic message