MenuTransition.swift

import UIKit

class MenuTransition: NSObject, UIViewControllerAnimatedTransitioning {
    
    private var state: TransitionState = .present
    private typealias Screens = (from: UIViewController, to: UIViewController)
    
    func presentationController(forPresented presented: UIViewController, presenting: UIViewController?, source: UIViewController) -> UIPresentationController? {
        return PresentationController(presentedViewController: presented, presenting: presenting)
    }
    
    func transitionDuration(using transitionContext: UIViewControllerContextTransitioning?) -> TimeInterval {
        return 0.3
    }
    
    func animateTransition(using transitionContext: UIViewControllerContextTransitioning) {
        let container = transitionContext.containerView
        let duration = transitionDuration(using: transitionContext)
        let screens: Screens = (transitionContext.viewController(forKey: .from)!, transitionContext.viewController(forKey: .to)!)
        switch state {
        case .present:
            present(duration: duration, container: container, screens: screens, transitionContext: transitionContext)
        case .dismiss:
            dismiss(duration: duration, container: container, screens: screens, transitionContext: transitionContext)
        }
    }
    
}

extension MenuTransition: UIViewControllerTransitioningDelegate {
    
    func animationController(forPresented presented: UIViewController, presenting: UIViewController, source: UIViewController) -> UIViewControllerAnimatedTransitioning? {
        state = .present
        return self
    }
    
    func animationController(forDismissed dismissed: UIViewController) -> UIViewControllerAnimatedTransitioning? {
        state = .dismiss
        return self
    }
    
}

extension MenuTransition {
    
    private func present(duration: TimeInterval, container: UIView, screens: Screens, transitionContext: UIViewControllerContextTransitioning) {
        container.insertSubview(screens.to.view, aboveSubview: screens.from.view)
        screens.to.view.transform = CGAffineTransform(translationX: -container.bounds.width, y: 0)
        UIView.animate(withDuration: duration, animations: {
            screens.to.view.transform = CGAffineTransform.identity
        }) { (_) in
            transitionContext.completeTransition(!transitionContext.transitionWasCancelled)
        }
    }
    
    private func dismiss(duration: TimeInterval, container: UIView, screens: Screens, transitionContext: UIViewControllerContextTransitioning) {
        UIView.animate(withDuration: duration, animations: {
            screens.from.view.transform = CGAffineTransform(translationX: -container.bounds.width, y: 0)
        }) { (_) in
            transitionContext.completeTransition(!transitionContext.transitionWasCancelled)
        }
    }
    
}

PresentationController.swift

import UIKit

class PresentationController: UIPresentationController {
    
    private var chrome: UIView = UIView()
    override var frameOfPresentedViewInContainerView: CGRect {
        return CGRect(x: 0, y: 0, width: containerView!.bounds.width-56, height: containerView!.bounds.height)
    }
    
    override init(presentedViewController: UIViewController, presenting presentingViewController: UIViewController?) {
        super.init(presentedViewController: presentedViewController, presenting: presentingViewController)
        chrome.backgroundColor = UIColor(white: 0.0, alpha: 0.5)
        chrome.alpha = 0.0
        chrome.addGestureRecognizer(
            UITapGestureRecognizer(target: self, action: #selector(tapGesture(sender:)))
        )
    }
    
    override func containerViewWillLayoutSubviews() {
        chrome.frame = containerView!.bounds
        presentedView?.frame = frameOfPresentedViewInContainerView
    }
    
    override func presentationTransitionWillBegin() {
        containerView?.insertSubview(chrome, at: 0)
        presentedViewController
            .transitionCoordinator?
            .animate(alongsideTransition: { (_) in
                self.chrome.alpha = 1.0
            }, completion: nil)
    }
    
    override func dismissalTransitionWillBegin() {
        presentedViewController
            .transitionCoordinator?
            .animate(alongsideTransition: { (_) in
                self.chrome.alpha = 0.0
            }, completion: nil)
    }
    
    @objc private func tapGesture(sender: UIGestureRecognizer) {
        presentedViewController.dismiss(animated: true, completion: nil)
    }
    
}

How to use

let menuTransition = MenuTransition()
let nv = UINavigationController(rootViewController: UIViewController())
nv.modalPresentationStyle = .custom
nv.transitionDelegate = menuTransition
present(nv, animated: true, completion: nil)