When indexing content from your WordPress posts, the Swiftype Search WordPress plugin will strip shortcodes, but retain the content inside the shortcode.
For example, a shortcode like
[youtube url=ZUD3qUtT3go] will be removed entirely, while a shortcode like
[column_left] This is the column [/column_left] will index the text "This is the content".
Of note, is that if your page has nested shortcodes, only the outer layer codes will be stripped out. Because of this you could potentially have shortcodes appear within your page documents.
You could potentially modify the plug-in to enhance this behavior. If you happen to go down that path, please submit a pull request!