I'm not sure this question has any kind of assessable answer, but one common assertion that seems to hold some water is the notion that numbers in discourse are distributed roughly logarithmically; this is the basis behind such patterns as Benford's Law. Granted, that usually applies to real numbers (ignoring sign), but it doesn't take too much work to apply it to the natural numbers.
Of course, the sum of the logs (or the integral of the log function) is unbounded, so using that to directly arrive at a probability distribution won't work. Ultimately, we will have to set an upper limit to the numbers we consider. We might use any of a number of very large numbers used in various proofs, but since you mention "daily life" and not "abstruse mathematics," I think it's reasonable to set an upper limit at about a trillion, the largest number that most people are likely to come across in everyday discourse (in connection with government budgets, let's say).
I would also aggregate all numbers with the same first three or so significant digits; that is to say, the number $345217469$ would be assigned a probability of essentially zero, as would most nine-digit numbers beginning $345$, but $345000000$ would be given the sum of all of their probabilities.
That's not quite ideal, but given that the logarithmic distribution is already an idealization (!), that's probably close enough to get in the neighborhood.