If you have the Plus plugin, it is supposed to share the page type sidebar setting with the 404 page.
BUT it looks like there is a bug.
@weaver, It seems the Search page results is not inheriting the Sidebar layout set for the 404 page. Is that a bug ?
The weird thing is on my test site the search page has no sidebar even if I select a sidebar layout for ALL page types. Yet the OP says he has a sidebar on his search page.
@Nickde Can you provide a link to your site