Satellites are routinely monitored by the likes of US, Russia and China. China for example is known to blind American spy satellites with lasers.
The goal is to track the approx 23000 (sources vary from 10 to 25000) objects, which could be anything from a satellite, debris from launches and even lost items from EVAs.
US monitoring systems
USSSN - US space surveillance network
Monitoring is accomplished in the US by the USSSN, the US space surveillance network. The majority of sensors are radars. They allow tracking of smaller objects, that cannot be tracked optically, they are distributed around the globe, they can track under any weather conditions. Facilities of the USSSN are located everywhere around the world, Alaska, California, Spain, Norway, Greenland, UK, and various islands in the atlantic, indian and pacific ocean. The objective is cpmplete coverage of the geosynchronous belt. There are dedicated sensors as well as civilian sensors that share their data. Optical sensors operate in favorable weather at night or during sun set/rise.
The organizational unit responsible is the US Strategic Command's Joint Functional Component Command for Space.
GEODSS - Ground based electro-optical deep space surveillance
is a dedicated optical system and is part of the USSSN. They can detect 20cm objects in geo orbit, which is not bad.
The telescopes can either track the stars, then the stars appear as dots and orbiting objects appear as streaks. This is SOP. The complementary mode is target track, which means, that the telescope tracks a space object, all other objects, such as stars, appear as streaks.
Radar is just radar. Depending on the system the output is either a data stream of tracks, video or pictures.
The tracking information of all these sensors is sent in realtime to Cheyenne Mountain Air Station, Colorado, where they.
The list of observing facilities in the screenshot is identical to the USSSN locations. Clear is Clear AFB in Alaska. Xuanhua is a chinese phased array radar on a mountain in northern china. The RUS sensors are from the SKKP.
US space catalogs
All the data is fed into the US space catalog, which tracks everything from small debris left over from space walks to the space station. The Air Force and the Navy maintain separate catalogs.
NASA publishes all non-classified objects. The offical catalog is available at https://www.space-track.org/
Classified tracks are not available, but it will come as no surprise that this information is readily available on the internet as this seems to be a rather popular hobby in astronomy circles.
A reasonably proficient hobbyist can track satellites with an optical telescope
Tracking is not as complicated as it may sound at first. Two observations are usually enough to determine the track. Usually everything stays in the same orbit or as a decaying orbit, if it is in a low orbit. Some satellites can maneuver and are therefore harder to track, but far from impossible.
Tracking every piece in orbit is very important to ensure safe access to space and safe operation in space. The ISS had to make maneuvers to avoid debris in the past.
Russia operates a similar system to the USSSN named SKKP. No further information is available
The Chinese satellite telemetry tracking and control network consists of the Xian satellite control center (XSCC), a number of fixed posts and three mobile units under the command of XSCC and ocean going instrumentation ships.
More information can be found at Wikipedia site for USSSN