Hi Gal, this depends on how you setup your security. When you say: 'the current user would be shown to everyone’, what do you mean? Shown for all Administrators? Or another user with a different user role? The problem you face, I guess, is because Account is a specialization of the System.User entity and by default it's setup so that you can only manage your own Account. Some solutions, without knowing the answer to my questions:
1. In app security: make sure you select the right User Role in User management (see screenshot below). For example: if a Manager/Administrator should be able to manage all/certain Employees (I don't know your situation), the Employee button needs to be selected.
2. In EmailConnector or EmailMessage entity add a string attribute ‘User’ or something. Then, whenever creating this object, fill this attribute by retrieving the current user and use a change object activity for EmailConnector/EmailMessage and fill that attribute. This way you are not restricted by security.