The issue you have is that if you use a "Random number generator" and you don't know how random that generator is, then the probability of collision is actually unknown. If the random number generators are correlated in some way, the probability of collision may dramatically increase - possibly many, many orders or magnitude.
Even if you have a very small probability of collision, you have a fundamental problem: The probability is NOT 0. This means that a collision WILL eventually occur, they just won't occur very often.
The more frequently you generate and use the UUIDs the sooner that collision is likely to be seen. (generating 1 a year means a longer waiting time than generating a million per second, all other things being equal).
If that probability is finite, unknown, and you use a lot of UUIDs then you need to consider the consequences of a collision. If it is not acceptable to throw an exception and shut down a business application, then don't do it! (Examples off the top of my head: "It's OK to shut down the web server in the middle of updating a library checkin... it won't happen often" and "It's ok to shut down the payroll system in the middle of doing the pay run". These decisions may be career limiting moves.)
You may have a worse case though, again depending on your application. If you test for presence of a UUID (ie, do a lookup) and then make a new one if one is not already there - which is a common enough kind of thing to do - then you may find you are linking records or making relationships, when in fact you are hooking up 2 things via a UUID that should not be hooked up. This is something where throwing an exception won't solve anything and you have an undetectable mess created somewhere. This is the kind of thing that leads to information leakage and can be very embarrassing. (ex: Log in to your bank and find you can see the balance of somebody elses account! Bad!)
Summary: you need to consider the way your UUIDs are used, and the consequences of a collision. This determines if you should take care to detect and avoid collisions, take some simple action in the event of a collision, or do nothing. A simple, single, one-fits-all solution is likely to be inappropriate in some circumstances.