In my reply I was assuming the use of an analog scanner or ham radio which have a limited bandwidth around any selected frequency. Any received spread spectrum signal received in that bandwidth will sound like static. That is pretty cool you can see the satellite transmission on your SDR display, are you sure its actually spread spectrum vs a broad bandwidth high data rate signal? I couldn't find much on HRIT transmission specs in a quick search.
Re the carrier and data separation (demodulation) question, the only way that could be done is if you know the spreading sequence code and have a receiver designed for it. If you look at a direct sequence spread spectrum signal you won't be able to see any carrier, what you would see is a higher "noise" level over the spread spectrum transmission bandwidth. Here is a "back of the envelope" explination that is approximate enough. The reason the signal looks like noise is the carrier is modulated by a very high speed pseudo random code (PRC) of "chips", which mathametically resembles white noise, and that is what spreads the signal out...the faster the chip rate the wider the signal spread and the lower the power level at any discrete frequency. The digital intelligence in the signal, which is sent at a data rate far lower than PRC chip rate, is combined with the PRC in a mixer circuit and then sent to the transmitter. To extract the intelligence the received signal is processed in reverse using the same time synchronized PRC (think of encoding as addition and decoding as subtraction, if you know what the add and subtract number is at any given time you can recover the intelligence). Because the PRC rate is much faster than the data rate, "pieces" of the received SS signal for a bit of information/intelligence (0 or 1) can be missing or corrupt and the correlator circuit can still decode it, i.e. if 60% of the PRC chips for a given data bit of intelligence are present or correct the correlator would assume what the bit was even though 40% of the chips were lost. That ability to loose PRC chips and still extract the intelligence is what gives spread spectrum processing gain and allows some incredible ranges at low transmit power (there are also other error correction protocols built in too that are not part of SS). So multiple spread spectrum radios can share the same center frequency and bandwidth because they will be using different PRC sequences. The way the codes are constructed a SS receiver will see all other received SS signals as background noise while its decoding the intelligence using the PRC of the transmitter. The older Verizon and Sprint cell phones used this type of system. Hopefully this explination is not as clear as mud
If you want to learn more just search on direct sequence spread spectrum.