The chances of someone randomly guessing a v-4 UUID is infinitesimally small. It's so tiny, it's not worth serious consideration.
Take a look at the math.
The following equation shows the number of guesses (n) required to generate a 50% probability of a correct match is 2.71 quintillion.
n ≈ 0.5+sqrt(0.25+2*ln2*(2^122))
≈ 2.71 × 10^18
Let's increase the probability (p) of a match to one in a billion.
Let: p = 1/(1 billion) = 0.000000001
UUID ≈ sqrt(2*2^122*ln(1/(1-p)))
≈ 103.1 × 10^12
I.e., to would take about 103 trillion guesses.
Given these odds, and the compute power required to engineer the hack, the likelihood of someone randomly accessing the UUIDs associated is so low, it just wouldn't be worth losing any sleepover.
Wikipedia has an in-depth description of the model and the use of UUIDs in database keys.
Always assume that any addressable network is hackable. But since your files will be effectively "unlisted" and semi-private using the UUID model should be sufficient to make them virtually inaccessible.