I have two responses to this question. Initially, I would have said that you should start at least attempting to figure out songs from the very beginning of your musical education. Even if you are just picking out "Twinkle, Twinkle" and "You are My Sunshine" on a single string.
I think that this is probably the ideal, but I have begun to wonder whether it is reasonable. I have met lots of beginners who may know the words to simple songs, but they do not know the actual tune. They cannot tell me if the second note/word in "You are my Sunshine" goes up in pitch, down in pitch, or stays the same. They do not yet understand the basic shape of the tune.
I therefore think that there should be an step before trying to blindly learn songs by ear. First learn a very simple tune in a simple key, C or G. Then transcribe that tune and play it in another key. You will already know the basic shape and the intervals between notes, but you still begin to work on ear training and playing what you hear. Once you can do this basic exercise with confidence, you are probably ready to try to pick songs out totally by ear.
And to answer your question, my experience is it always takes people longer than they are willing to admit, and there are lots of people who play guitar (notice that I do not call them musicians) who never learned to do it. Ear training is hard, frustrating, and all to easy to overlook. On the other hand, it pays off down the road.
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