Short answer: In the vast majority of COVID-19 cases, an infected person will develop antibodies that are effective at preventing reinfection or transmission; however, there have been cases where a previously infected person has tested positive months later. Bottom line, even if you have antibodies, take the same precautions you would normally take.
Longer answer: The human immune system is extremely complicated and varied, but one of the most important components are disease-specific antibodies. These proteins are formed by white blood cells when foreign antigens are detected. However, since diseases have different cell structures and binding receptors, each antibody has to be tailored for each disease. Therefore, an antibody for one coronavirus most likely isn’t effective against COVID-19. Another factor is the unpredictability of viral genetics; extremely infectious diseases like COVID-19 pass through an inordinate amount of hosts, both human and animal, which gives the virus plenty of opportunities to develop mutations. These mutations could be inconsequential for the immune system response, but a change to the viral receptors could completely reset any immunity. There’s already evidence that the virus has changed significantly since it was first identified in China; it seems that the virus that emerged from Europe is remarkably more infectious. Variants have also popped in Brazil, the United Kingdom, and South Africa with slightly different receptors, making them potentially capable of thwarting vaccines and antibodies. Given COVID-19’s newness, it is wise to continue to take precautions against any infection even if you have antibodies.