Yes. Your exchanges in Messages are private between you and the other person. They are not public and appear nowhere on your profile.
A few useful points to know:
- Chat offers automatic translation to talk with members in other languages: you can switch between the translated text and the original.
- You can share photos in a conversation; only share what you are comfortable showing.
- If an exchange makes you uneasy, you can report or block the person at any time.
For your safety, never share passwords, banking details or codes received by text. Remember there is no payment inside the app: any request for "secure payment", a deposit or delivery fees via a link is a sign of a scam to report.