I regret you cannot with 'native' functionality (ie not a script). Support has:
If an editor duplicates the protected sheet, copies the workbook, or uploads a new version, sheet protection will be disabled. Also, if a viewer creates a copy of the spreadsheet, the data on the protected sheet can be edited in the copy.
And this warning:
Protected sheets and ranges can help prevent people from unintentionally modifying content in a spreadsheet, but they should not be used for security purposes because not all actions are blocked (for example, printing, copying and pasting, and importing and exporting). Only share spreadsheets with people you trust.
Protected ranges are sheet specific. You may have noticed that when protecting ranges, even if only supplying range references, the relevant sheet name is prepended to these automatically. You might sacrifice those in your 'existing' sheet (say
S1) for the sake of your copied sheet (say
S2) by manually changing all the
S2! in the Protected sheets and ranges window.