What if I don't want to share contact details of my current employer, because they are not aware that I might leave the company soon?

You can fill in that a certain work experience is at your current employer, you are not required to share this data. There are some exceptions, some clients require this input. If you cannot select the option that it is your current employer, you are required to fill in contact details. If you have issues with this, please contact your future employer.