Same goes for registering a LLC with the Indiana Secretary of State , in fact you don't even need to provide a physical location address there either and is much cheaper annually than DE.
In regards to doing business in your State and what was mentioned above, you can simply for the same cost as filing for a foreign LLC in your State, form another LLC there where your DE/IN LLC is the managing member. That way your information will never become public and be seen by anyone. It will only show membership online by your DE/IN LLC and when someone goes to the SOS website in DE/IN, it won't show your info there either.
Therefore your info is hidden in DE/IN and since that LLC is the only Managing Member in your States LLC the path will lead to nowhere.
One is the Holding Company, and the other is the Subsidiary. Since they are LLC's you are not double taxed and do not need to file individual returns per LLC, it all falls through you in the end avoiding double taxation.