Early decision policy does not distinguish private vs state

You can apply to any other non-binding colleges while your early decision is underway. Once you are offered the early decision, you have two to three weeks to decide if you want to accept the offer or not.
 

Once you decide to accept the offer, you are required to withdraw all other college application, regardless state or private, there is no distinction.

If you reject the offer and you are released the agreement, that is another story.

High school counselor is required to sign the agreement form. 

To withdraw the agreement after regular decision has been released is rare and as such, it will be noted by the college and by high school counselor.

We live in a big data era.