Replace Action\Action in a controller with both Get and Post requests

You cannot call a parent private method from the child class: $this->getDataPersistor().

In this case, if you use the <preference> way to override this class, you need to re-declare getDataPersistor() method.
However, I strongly recommend we to use the Plugin way for execute method as we can.

Try following code:

namespace Pos\Contact\Plugin;


class Post
{
    /**
     * @var \Magento\Framework\Controller\Result\RedirectFactory
     */
    protected $resultRedirectFactory;

    /**
     * @var \Magento\Framework\Message\ManagerInterface
     */
    protected $messageManager;

    /**
     * @var \Magento\Framework\Translate\Inline\StateInterface
     */
    protected $inlineTranslation;

    /**
     * @var \Magento\Framework\Mail\Template\TransportBuilder
     */
    protected $transportBuilder;

    /**
     * Core store config
     *
     * @var \Magento\Framework\App\Config\ScopeConfigInterface
     */
    protected $scopeConfig;

    /**
     * Post constructor.
     * 
     * @param \Magento\Framework\Controller\Result\RedirectFactory $resultRedirectFactory
     * @param \Magento\Framework\Message\ManagerInterface $messageManager
     * @param \Magento\Framework\Translate\Inline\StateInterface $inlineTranslation
     * @param \Magento\Framework\Mail\Template\TransportBuilder $transportBuilder
     * @param \Magento\Framework\App\Config\ScopeConfigInterface $scopeConfig
     */
    public function __construct(
        \Magento\Framework\Controller\Result\RedirectFactory $resultRedirectFactory,
        \Magento\Framework\Message\ManagerInterface $messageManager,
        \Magento\Framework\Translate\Inline\StateInterface $inlineTranslation,
        \Magento\Framework\Mail\Template\TransportBuilder $transportBuilder,
        \Magento\Framework\App\Config\ScopeConfigInterface $scopeConfig
    ) {
        $this->resultRedirectFactory = $resultRedirectFactory;
        $this->messageManager = $messageManager;
        $this->inlineTranslation = $inlineTranslation;
        $this->transportBuilder = $transportBuilder;
        $this->scopeConfig = $scopeConfig;
    }

    /**
     * @param \Magento\Contact\Controller\Index\Post $subject
     * @return $this|void
     */
    public function aroundExecute(
        \Magento\Contact\Controller\Index\Post $subject
    ) {

        $post = $subject->getRequest()->getPostValue();
        if (!$post) {
            return $this->resultRedirectFactory->create()->setPath('*/*/');
        }
        $this->inlineTranslation->suspend();
        try {
            $postObject = new \Magento\Framework\DataObject();
            $postObject->setData($post);

            $error = false;

            if (!\Zend_Validate::is(trim($post['name']), 'NotEmpty')) {
                $error = true;
            }
            if (!\Zend_Validate::is(trim($post['comment']), 'NotEmpty')) {
                $error = true;
            }
            if (!\Zend_Validate::is(trim($post['email']), 'EmailAddress')) {
                $error = true;
            }
            if (\Zend_Validate::is(trim($post['hideit']), 'NotEmpty')) {
                $error = true;
            }
            if ($error) {
                throw new \Exception();
            }

            $storeScope = \Magento\Store\Model\ScopeInterface::SCOPE_STORE;
            $transport = $this->transportBuilder
                ->setTemplateIdentifier($this->scopeConfig->getValue(\Magento\Contact\Controller\Index\Post::XML_PATH_EMAIL_TEMPLATE, $storeScope))
                ->setTemplateOptions(
                    [
                        'area' => \Magento\Framework\App\Area::AREA_FRONTEND,
                        'store' => \Magento\Store\Model\Store::DEFAULT_STORE_ID,
                    ]
                )
                ->setTemplateVars(['data' => $postObject])
                ->setFrom($this->scopeConfig->getValue(\Magento\Contact\Controller\Index\Post::XML_PATH_EMAIL_SENDER, $storeScope))
                ->addTo($this->scopeConfig->getValue(\Magento\Contact\Controller\Index\Post::XML_PATH_EMAIL_RECIPIENT, $storeScope))
                ->setReplyTo($post['email'])
                ->getTransport();

            $transport->sendMessage();
            $this->inlineTranslation->resume();
            $this->messageManager->addSuccess(
                __('Thanks for contacting us with your comments and questions. We\'ll respond to you very soon.')
            );
            return $this->resultRedirectFactory->create()->setPath('contact/index');
        } catch (\Exception $e) {
            $this->inlineTranslation->resume();
            $this->messageManager->addError(
                __('We can\'t process your request right now. Sorry, that\'s all we know.')
            );
            return $this->resultRedirectFactory->create()->setPath('contact/index');
        }
    }
}