The main people who oversee this (at least in the US) is the Joint Space Operations Center (JSpOC), From Wikipedia:
The purpose of the JSpOC is to provide a focal point for the operational employment of worldwide joint space forces and enable the Commander of JFCC SPACE (CDR JFCC SPACE) to integrate space power into global military operations. The JSpOC is located at Vandenberg Air Force Base in California and is the organization responsible for performing all of the orbit determination activity necessary to maintain the US space catalog.
One of the primary missions of JSpOC is to track everything in space, using the Space Surveillance Network (SSN), a worldwide network of 30 space surveillance sensors. They advertise that they can reliably track objects down to 10cm in size (it is postulated that they have the capability to track smaller objects). Part of their tracking mission is to inform satellite operators and launch providers about possible collisions from vehicles to be launched as well as every day as debris and satellites circle the earth.
Per the USSTRATCOM page on the JSpOC:
On a routine basis, the JSpOC conducts conjunction analysis for all active spacecraft. During human space flight launches, the center computes possible close approaches of other orbiting objects with the flight path of the Soyuz and the International Space Station (ISS). The JSpOC constructs a theoretical box around a high-interest object, (e.g. the ISS) and projects the flight path several days in advance of the launch. If any of the cataloged objects intersect this theoretical box, the JSpOC forwards the analysis to NASA. NASA makes the determination whether or not to change the flight path of the ISS. NASA offers to the general public, on its website, the opportunity to track various satellites.
In the past, there has been only one collision between two satellite in orbit that I am aware of, it is referred to as the 2009 satellite collision it was between Iridium 33 and Kosmos-2251, a US civilian communications satellite and a Russian military communications satellite. It released a ton of debris into orbit as the hit one another at 42,120 km/h.
Here is an image of the tracked debris cloud after 50 minutes:
Rlandmann CC BY-SA 3.0