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First free your mind of being stuck in the frequency domain.
It is true that ss has wider bandwidth. The total energy per bit is approximately the same, so there is a smaller energy density per unit of spectrum for ss. This can be very advantagous if you are trying to become 'invisible' on a spectrum analyser for stealth transmission.
For CDMA communication the original data is diced into a bunch of small bits called chips by a psuedo-random code. This random sequence of high bit rate chips is used to modulate the RF, usually phase shift keying. The spectrum is broad and can actually be lower in energy per Hz then normal background blackbody radiation noise when viewed at a reasonably usuable distance from transmitter.
To demodulate the CDMA the original psuedo code is cross correlated to incoming signal. When it aligns in time and phase the original data appears out of the noise. For cellular, different psuedo codes are used for different users so multiple data channels can share the same spectrum. Using the same RF 'channel' for multiple user transmission is hard for some folks to swallow, there so my starting comment about freeing your mind from RF channel frequency spectrum domain thought.
There is a second type of spread spectrum called frequency hopping. This is like conventional narrow band communication but there is a psuedo random channel changing sequence so receiver can match the hopping sequence. Error correcting algorythms can be employed on data so if one of the channel hops happens to get interfered with and lost the redundancy of data in other hops will provide correction to recover original data.
The psuedo random sequences are carefully selected to have equal 'Ones and Zeros' and minimum amount of cross correlation across the original data bit time. It is like breaking in half, a dozen egg shells and fitting the matching halves back together again after they were mixed up. You want the jagged edge pattern to be different between any other of the egg shells so only the original halves match up again. The original two half shells from a single egg 'correlate' together. You must try the fit by rotating the two egg shell halves around up to 360 degrees rotation to test for a fit. This is similar to the time alignment syncronization process for the psuedo random sequence of chip bits.
The recovered output signal to noise ratio obeys the original data bandwidth, not the spread spectrum bandwidth. For GPS the original data bandwidth for the 50 bps data is only 25 Hz so the signal can be recovered from a very weak signal. The high chip rate (1.023 million chips per second) bit width time is used for precise timing by using the fact that correlation of the pseudo random sequence produces no significant output until the correlation gets close to single chip width time sync.