We don't know, and may never know or won't know for a long time.
The reason we think the recently approved mRNA vaccines are effective is due to randomized controlled trials. You take a population of people who haven't previously gotten COVID-19, give half the vaccine and half the placebo, and determine efficacy based on the ratio of subsequent infections in both groups.
To answer your question, whether there is a benefit to vaccination after having COVID-19, you'd have to repeat the same trial in people who were already infected. That's...probably not going to happen. For one, any protection that prior infection provides is going to mean the "control" group in such a trial has a lower infection rate. This means that the overall power of the study will be lower for the same vaccine protection, so they trial may need to be much much larger to show an effect. Trials are expensive, and there's little reason to do this trial so little chance anyone will want to fund it. Companies manufacturing the vaccine want to show it works in the uninfected population, because most people still remain uninfected. Secondly, to answer conclusively that the vaccine has no benefit in people previously infected is even more difficult; it's actually statistically impossible to demonstrate this so it's necessary to set a threshold>0 to compare to. It's very unusual to design trials like this for anything but safety.
Overall, it's likely that people who have had COVID-19 before will still be recommended to get vaccinated. We won't have sufficient data to recommend against it, and can't be certain those people won't benefit from the vaccine. Prior knowledge of other coronaviruses causing "common cold" illnesses suggests that coronavirus immunity is not long-lasting; the worry is that people with generally cold-like symptoms will only get cold-like immunity to the virus, and hope that the vaccines will elicit a stronger immune response. We still don't know exactly how long vaccine protection will last (because the vaccines are still so new), but it's far more likely for us to understand the extent of vaccine protection than to understand the effect of natural infection since the former is possible in a controlled study setting.