The short answer is – we don’t know, but it isn’t looking good. We simply do not know enough yet about this novel virus. Pertinent questions include: how effective is the immunity that results from infection and (this is critical) how long does it last? How much of an infection is necessary to trigger immunity? Will asymptomatic cases be immune? And what level of immunity in the population is necessary to achieve herd immunity, given this varies for different viruses depending on how contagious and infectious it is?
We don’t really know the answer to any of these questions, but the early clues are not encouraging. Coronaviruses in general tend not to provoke lasting immunity (years), and SARS-CoV-2 may be no different. One recent study published in The Lancet, for example, found that, if you standardize to the general US population, about 9.3% of people have antibodies to SARS-CoV-2. There were regional differences that follow the density of COVID cases, with the highest regional percentage being about 25% in the Northeast (and about 35% in NY state). This number is about 10 times the number of confirmed COVID cases, so many people are being exposed to the virus without ever being diagnosed (whether or not they develop symptoms).
This number is low compared to what is needed for herd immunity, which varies from 50-90% depending on the virus. Even for the low end of the spectrum, we would need five times as many people exposed as we have now. That would probably mean about 1 million Americans dead just to reach the low end of the herd immunity range. That number could easily have to double, as COVID is highly contagious.
But also, we still don’t know how long immunity lasts. We know it lasts at least 3 months, but that is not very long.