How do we separate source signals in spread spectrum?
There are (at least) two types of spread-spectrum.

Frequency hopping and direct sequence spread spectrum (DSSS)

In DSSS all transmitters share the same frequency but with a different spreading code.

Multiple transmitters do interfere but after doing a cross-correlation using the specific code the unwanted signals get reduced by the cross-correlation factor. Ideally there will be zero cross-correlation but in practice, the codes used are not perfect and other effects such as doppler and range attenuation will make things worse.