When configuring an Objects Search Layout you have the option to "Override the search result column customizations for all users". I have been unable to find any Documentation that describes how a User could personalize their Search Layout, nor can I find any User Setting that allows such personalization. The only reference I can find is on the Documentation on how an admin can Configure the Search Layout at the Org level. https://help.salesforce.com/apex/HTViewHelpDoc?id=customize_searchlayout.htm&language=en_us
I don't see this in lightning but in the classic, it is available once you search something and dig into a particular object in the left sidebar and then you will see a customize button.