Free time is a musical meter with no time signature, many pieces like the Concord Sonata and the Gnossienne No. 1 has no time signatures. But still they have note duration. I don't understand, because when there's no time signature, shouldn't there also be no note duration? If so, how do I notate note duration in free time music?
"Meter" and "note duration" are mostly orthogonal concepts. You can still have the latter entirely without the former. You do need note durations to properly represent a meter, but the meter doesn't actively dictate much about how long your notes actually last.
It's better to think of free time as more about style and emotion. Borrow some time here, slow down there, speed up again here with the crescendo. But you're still working within a musical framework. The song isn't just a series of pitches with no temporal information at all. You could, of course, write some music that way if you wanted (and there are certainly such existing pieces), but that's not what is typically done.
While the tempo/timing/feel might vary wildly throughout the piece, and even measure to measure sometimes, it should not vary to an extreme between individual notes. The timing relation from one note to the next is a large part of what makes a piece of music recognizable across what may otherwise be very different performances.
Nonstandard notation almost always is trying to communicate something beyond "play this note for this long at this time"; the Concord uses barlines as phrasing indications; the Gnossienne uses none at all to indicate that the performance should be one flowing stream of continuous sound.
For performance, consider the durations as relative rather than specific. Assuming that the piece is also meant to be played rubato as Matthew suggests, a notated half-note is about twice the length of a notated quarter-note, and half the length of a notated whole note.
It's somewhat like when you read a text out loud. There's nothing to tell you which words you have to read quickly, or slowly. You feel your way through, well or badly. But here, the relative speed of the notes is suggetested. It doesn't have to be rigidly adhered to, but it's a guide.
Short answer: the notes still need durations, otherwise the performer won't know how long to play them for.