How to access protected property in feed object?
There is no public method on the State object that returns the messages. The public methods on the object are:


progress()
setCompleted()
setMessage()
displayMessages()
logMessages()


The method displayMessages() is the closest method you can use to access the messages, the messages are then transferred to \Drupal\Core\Messenger\Messenger, which has a method called all() to return all the messages on that object.

PHP also has an API called 'Reflection', which - among more - allows you to bypass visibility settings on class properties and methods. The main purpose for this API is to retrieve doc comments for functions, classes and methods. I wouldn't recommend to use this unless if it is for debug purposes.

Note that since Feeds 8.x-3.0-alpha6 the messages are also displayed on the feed page while an import is not yet completed.

Why do you need access to the messages? Maybe you can open a feature request ticket in the Feeds issue queue to request adding a public method for retrieving the messages.