If you are designing your own employee engagement scale, then see the link to SurveyMonkey below for designing a survey with the questions, that are most relevant to your research question, for  both your private and public sector sample (between-subjects - same questionnaire design).

I suggest you go to Google Scholar (see link below) and search for 'employee survey scale engagement' and either use a survey that has been published in a journal article relevant to your field, or use a mixture of questions from different scales relevant to your research question, controlling for likert scale similarity etc.

See also Alan Bryman's methodology readings if you are developing your own scale.

https://www.surveymonkey.co.uk/mp/employee-engagement-survey/

https://scholar.google.co.uk/scholar?hl=en&q=employee+engagement+survey&btnG=&as_sdt=1%2C5&as_sdtp=&oq=employee+eng

https://books.google.co.uk/books?id=l7u6BwAAQBAJ&printsec=frontcover&dq=alan+bryman&hl=en&sa=X&ved=0ahUKEwituLf_xqvVAhWCJMAKHThUDQYQ6AEILTAB#v=onepage&q=alan%20bryman&f=false

https://books.google.co.uk/books?id=N2zQCgAAQBAJ&printsec=frontcover&dq=alan+bryman&hl=en&sa=X&ved=0ahUKEwituLf_xqvVAhWCJMAKHThUDQYQ6AEIJzAA#v=onepage&q=alan%20bryman&f=false

Thank you a lot dear Michelle. I found some questionnaires in journal articles. I think I ll use mixture of questions from different questionnaires to operationalize constructs from my research model.