Currently, NORAD and others use large radar arrays to keep track of objects. Simplified trajectories are what appear in the common TLE databases.
Conjunction analysis is done by propagating the trajectories of these objects. There are many types of model used here, but broadly speaking they model the conjunction of large error boxes (ellipsoidal or rectangular) around each object. These are typically many kilometers across. Many of the propagation models are proprietary.
There is a gap from launch conjunction (airspace / FAA) to space conjunction - there is a lag from when an object "appears" in space, until it appears on radar and its trajectory is tracked and propagated. For more detail, see here for example.
When there is the potential for conjunction, NORAD / JSpOC / SpaceTrack send notifications to individual satellite operators warning them of this, in the hope that they plan collision avoidance manoeuvres. Each operator will have their own tolerance of risk, and as in the case of the Iridium-Cosmos collision, their risk appetite was too large. I recall that Iridium were receiving 400 messages per day or per week indicating potential collisions.
Operators are likely to perform routine orbital manoeuvres at times that are less convenient, just to avoid debris. That way, the avoidance doesn't "cost" them anything apart from some rescheduling of planned operational manoeuvres. This is one of the main reasons the debris problem is getting worse - no-one has a financial incentive to do anything about it.
Yes, there are a lot of objects in orbit, but space is also very very large. LEO is about $10^{12} km^3$, so the overall density of objects and thus the probability of any collisions is overall "low" - it's definitely higher than I would like to personally see but is apparently acceptable to satellite operators.
![Density of debris in LEO](https://i.sstatic.net/uL6xq.jpg)