How does a diode "rectify" an alternating electrical signal from an antenna in a diode type crystal radio?
First: A crytstal radio works only with AM radio, i.e. the audio signal must be coded as the amplitude of the RF signal.

The diode does in deed cut the negative part of the RF signal, but those parts are very short and evened out by a capacitor after the diode (or just by the capacitance of the speaker).

That resulting signal is the amplitude of the RF signal and is the actual audio signal.

Since the radio frequency (carrier) is much higher (e.g. by factor 1000 or more) than the modulating signal (audio signal) the upper and lower halfes of the RF are "duplicates" (as you call it). So no information is lost by rectification.