If you like to have invisible columns in filters, that can be achieved by enabling “Filter in form” option in table edit page.
“Filter in form” option will create filters as a separate section from the table and additionally show filters for all hidden columns.
You can choose which column filter will be visible/hidden by unchecking/checking the Add a filter for this column button in the column settings for desired column.

For showing and hiding column filters is responsible “Advanced column filters ” option on the table settings page.
You can also hide each column filter in column settings by unchecking the “Add a filter for this column” button in Filtering tab for desired column.
By checking or unchecking this settings check box and saving, table will show or hide column filters.

In order to show column filters in header “Render advanced filter” option needs to be changed to “In the header” on wpDataTables Settings page.