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This issue occurs because Ubuntu 13.10 ships with PulseAudio 4.0 and Skype doesn't properly support it yet.

A. Install Skype from the Ubuntu 13.10 (Saucy Salamander) repositories, which comes with a fix for this issue. This is the recommended way of installing Skype in Ubuntu since it usually comes with extra fixes that aren't available in the deb downloaded from Skype's website.

Firstly you'll need to enable the "Canonical Partners" repository. To do this, launch "Software & Updates" from System Settings and on the Other Software tab, enable "Canonical Partners":

Then, update the software sources and install Skype:

sudo apt-get update
sudo apt-get install skype

B. If for whatever reason you don't want to install Skype from the Ubuntu repositories, you can apply a fix to the Skype version downloaded from its official website.

To fix the Skype no sound / distorted sound issue in Ubuntu 13.10, run the following command:

sudo sed -i 's/^Exec=.*/Exec=env PULSE_LATENCY_MSEC=30 skype %U/' /usr/share/applications/skype.desktop

The command above replaces "skype" from the Skype desktop file with "env PULSE_LATENCY_MSEC=30 skype", this being a work-around for getting Skype to work with PulseAudio 4.0. 

If you want to launch Skype from a script or add it to Startup Applications, use the following command: "env PULSE_LATENCY_MSEC=30 skype" instead of "skype".