(In reply to Sören Hentzschel from comment #0)
A search field for private browsing was implemented in bug 1508364. It's a great addition but it doesn't respect the user's setting for displaying a search field in the new tab page. There is no reason to assume that if the user disables the search field, the user wants to have one in private windows. I think it should be respected if the user makes an explicit choice.
New tab in private browsing mode isn't meant to be customizable at this time. Since PBM doesn't have the other sections of new tab (top sites, pocket stories, highlights) it should be treated as a special case instead of inheriting partial settings. We should maybe consider a special set of settings for PBM as people may want to customize one experience over the other.