I've been thinking for this for a while. It seems there's no ethical experiment to perform to check if immunity is actually developed (the obvious experiment would be to expose someone that have recovered from the infection, but that's unethical).
My idea is that lack of immunity would eventually result in observation of reinfected subjects and one should be able to estimate the likelyhood of that. An experiment would for example be that if we study a set of subjects over time and at one time 1% are infected and later when these have recovered we at a later time find that 1% are infected then perhaps 1% of these would be expected to be among those that were infected the first time (unless immunity has developed).
How did we realize that we get immune to measles, chickenpox and mumps for example?