One thing that is hard to quantify is the risk of long-term effects, which are unknown.
There is an elegant argument, due to Laplace, that says that if you have an urn containing red and blue balls, you extract N balls, and M of them are red, you should assume that the probability that the next ball is red is (M+1)/(N+2), and not M/N as one might naively assume. The general case requires integrating the beta function, which is kind of advanced, but the M=0 case can be done with elementary calculus, as follows.
Call X the probability of extracting a blue ball, which we view as a property of the urn. If we don't know anything about X, before we extract any balls, we should assume a uniform prior distribution P[X]=1 for 0<=X<=1 (this is the main and only assumption). The probability of seeing M=0 red balls after extracting N, for given X, is the same as the probability that all balls are blue, i.e., P[M=0|X]=X^N. But we care about P[X|M], not P[M|X]. By Bayes' theorem, P[X|M] is proportional to P[M|X]P[X], times a proportionality constant that makes the total probability be 1. Because we assumed P[X]=1, we have that P[X|M=0] is proportional to P[M=0|X]=X^N. The integral of X^N between X=0 and 1 is 1/(N+1), yielding P[X|M=0]=(N+1) X^N. The expected value of X is the integral for X=[0,1] of X P[X|M=0], which is E[X]=(N+1)/(N+2). This is the expected probability of a ball being blue, with 1-E[X]=1/(N+2) being the probability of a ball being red. QED.
Now say we have historically observed 1000 vaccines and they were all safe in the long term. It is still perfectly rational to assume that there is a 1/1002 chance that this vaccine is unsafe in the long term. Anybody claiming otherwise better have a cogent argument about why the prior probability should not be uniform. Saying that 1000 vaccines were long-term safe and thus this one is long-term safe is equivalent to assuming a prior of the form P[X]=1/(X (1-X)), which is hard to justify (and diverges at 0 and 1).
Basically, the problem is that we are entering the territory where the risk from the disease is comparable to a rational estimate of the risk of what we don't know, and it's hard to come to any kind of cogent conclusion.
But anybody who claims that all past vaccines were long-term safe and thus this one is long-term safe clearly does not understand basic probability.