The data you create and store on Verifiable is completely private, there is no public profile associated with your Verifiable account.
Your Verifiable account is a hub for you to store and manage your digital credentials. Only the applications that you connect to your Verifiable account will be able to access and update your data. You have complete control over which applications are connected. When you share your Verifiable data with other applications, they may create public profiles for you so it is important to always reference the terms on various applications so that you are aware of how your data is being treated.