Sec.62 of the Companies Act , 2013 and the Issue of Share and Debentures , Rules 2014 do not require the shares under rights issue or ESOP to be fully paid up. In case of a private limited company, also do check the Articles of Association for any such clauses.
Yes, the Private Company may issue partly paid up shares under Right Issue or ESOP, subject to the power provided in the Articles of Assocoation of the Company.