How unique is UUID if generated on devices and not servers?

This question answers it in a general sense and it doesn't specify on what will happen if the UUID generation is not centralized.

I recently saw an architecture where all the devices (1M+ ios and android) were generating UUID4 keys(using their own generator functions/library) and those keys when synced were marked unique on the server's database. I fear that since around a million devices will try to generate locally, the chances of collision will be higher than described in the question above.

I've used centralized UUIDs before but new to this type of distributed system, so I'm taking the analogy of a prime number generator, running in parallel on different environments, which will make the end result not prime/unique. Please correct me if my understanding is wrong.

Also, please share any good articles on gotcha's and using UUID in distributed environments.