If you are targeting ios 8 and above you can set the modal presentation style to "over current context" and you are done.
If ios 7 and below, you would have to create a custom transition style so that the presenting screen doesn't go blank after transition. That is rather complicated.
The solution I present offers a lot of flexibility: make a screenshot before showing the modal dialog and set that as the background image for the application window. By default, that background is black (that is what you see when the back view controller dissapears). Change the background to the screenshot of the app. Make the screenshot in the viewWillAppear or viewDidLoad method of your transparent view. This works even with push segues, not only modal dialogs, but you should avoid animations. In general, avoid animations which affect the position of the background view because those will make it seem like it snaps back into place when transition finishes. It is a good idea to reset the background to its previous black image on viewDidDissapear to avoid unwanted effects.
You can maintain a stack of such background images and you can do multiple "transparent" push seques. Or have some complex/deep menu which appears on top of some main screen. For these many reasons I think this solution is better than rolling your own transitioning code. It is more flexible and easier to implement, and you don't have to deal with the animations yourself.