You need to override the "message_post" method in crm.lead object. When anything gets post in the chatter, this method gets called.

Override this method, call the super method first so that it will create the record of "mail.message" and return the recordset of the mail.message record. Now, copy the mail.message and pass the default dict in copy method (msg.copy({'res_id': self.partner_id.id, 'model': 'res.partner'})).

This will post the same message from Lead to related Partner.

Ex:

    @api.multi
    @api.returns('mail.message', lambda value: value.id)
    def message_post(self, body='', subject=None,
                     message_type='notification', subtype=None,
                     parent_id=False, attachments=None,
                     notif_layout=False, add_sign=True, model_description=False,
                     mail_auto_delete=True, **kwargs):
        msg = super(CrmLead, self).message_post(body=body, subject=subject, message_type=message_type,
                     subtype=subtype, parent_id=parent_id, attachments=attachments,
                     notif_layout=notif_layout, add_sign=add_sign, model_description=model_description,
                     mail_auto_delete=mail_auto_delete, **kwargs)
            if self.partner_id:
                msg.copy(default={'res_id': self.partner_id.id, 'model': 'res.partner'})
        return msg