Can Signal access private posts online?

No.

The Signal is an open-source intelligence gathering aggregation platform and can only access publicly available information (information the user has agreed to be publicly viewable when they signed up for their social media account). We do not access any private, passworded or encrypted content.

If the information you wish to access is private - you will likely need a warrant/subpoena to see each post.

You cannot accidentally breach anyone's privacy using Signal.