It depends quite a bit on your definition of 'spying'. For example, the Ubuntu distribution as a default has "Yes send the systeminfo to Canonical", which means that Canonical collects information about your system (which you can turn off if you want to). A quote:
The data includes information about your PC’s hardware, including the manufacturer, BIOS version, and the model of your CPU. It also includes information about your software, such as the version of Ubuntu you installed, your chosen desktop environment, whether you’re using the Xorg or Wayland display server, and the options you chose while installing Ubuntu. Other information, such as your time zone, information about your partitions, and your display’s resolution is also sent.
According to Canonical’s Will Cooke, this data is designed to help Ubuntu learn how many users it has, what features they use, and what hardware they have, allowing Ubuntu’s developers to focus on what’s important. Ubuntu doesn’t keep the IP address associated with each report and has no way of tying it back to an individual PC.
If you use Gnome,
GNOME has its own set of spyware, for example geoclue2 tries to detect your geographical location and shares it with other software, iio-sensor-proxy gathers intelligence about your devices and hardware and shares that data with other software
The Gnome example illustrates that, although Linux may perhaps be spyware-free, other software may allow spying anyway. So be careful what you install.
If spying is being done, the data needs to be send out to the spy. If you monitor you network traffic, you should not see any unexpected traffic. But you'll need to monitor that closely. And the data streams may be encrypted, so you would probably need a proxy or firewall that breaks open the SSL session (MITM proxies). There are people doing this, actually, and they have not raised concerns (yet).
On the other hand, there are (despite the lack of real evidence) concerns that the Deepin distribution actually contains spyware, just because it is made in China.