Change private post viewing capability from Editor to anyone logged in.

So, like the title states, I’d like to change the level of who can view private posts from Editor, to anyone that is logged in. I’ve been through all of the plugins, and there is one that does this, but it also has some undesirable side effects. I tried to go through that plugin, and see if I could just use part of it, but couldn’t make it work. Any help is greatly appreciated.