Covid-19 as a recognized disease has not been around long enough to answer these questions. There was a tiny study of 4 rhesus macaque monkeys that showed that recovered monkeys (2) were able to resist a second inoculation of the same strain of SARS-CoV-2.
We also know that patients with mild respiratory Covid-19 can still shed virus after their symptoms have resolved, and these shed virus are potentially infectious.
"If you had mild respiratory symptoms from COVID-19 and were staying at home so as not to infect people, extend your quarantine for another two weeks after recovery to ensure that you don't infect other people," recommended corresponding author Lixin Xie, MD, professor, College of Pulmonary and Critical Care Medicine, Chinese PLA General Hospital, Beijing.
The authors had a special message for the medical community: "COVID-19 patients can be infectious even after their symptomatic recovery, so treat the asymptomatic/recently recovered patients as carefully as symptomatic patients."
Concerningly a recent report, not yet peer reviewed or replicated, has found a low level of immunity to Covid-19 in recovered patients
Researchers in Shanghai hope to determine whether some recovered coronavirus patients have a higher risk of reinfection after finding surprisingly low levels of Covid-19 antibodies in a number of people discharged from hospital.
A team from Fudan University analysed blood samples from 175 patients discharged from the Shanghai Public Health Clinical Centre and found that nearly a third had unexpectedly low levels of antibodies.
In some cases, antibodies could not be detected at all.
“Whether these patients were at high risk of rebound or reinfection should be explored in further studies,” the team wrote in preliminary research released on Monday on Medrxiv.org, an online platform for preprint papers.
Although the study was preliminary and not peer-reviewed, it was the world’s first systematic examination of antibody levels in patients who had recovered from Covid-19, the disease caused by the coronavirus, the researchers said.
All of the patients had recently recovered from mild symptoms of the disease and most of those with low antibody levels were young. The researchers excluded patients who had been admitted to intensive care units because many of them already had antibodies from donated blood plasma.
The researchers said they were surprised to find that the antibody “titer” value in about a third of the patients was less than 500, a level that might be too low to provide protection.
“About 30 per cent of patients failed to develop high titers of neutralising antibodies after Covid-19 infection. However, the disease duration of these patients compared to others was similar," they said.
The team also found that antibody levels rose with age, with people in the 60-85 age group displaying more than three times the amount of antibodies as people in the 15-39 age group.
So, preliminary data (not yet peer reviewed) suggests that the immune response is weak particularly in younger people and may not protect you against re-infection. In which case you can then become a vector to infect other people.
Various responses to this preliminary report include
Although the paper has not been officially peer-reviewed, scientists on Twitter identified a few possible caveats in the paper which might affect the conclusions. Firstly, some suggested that perhaps some of the patients who had low or undetectable levels of protective antibodies may not have had COVID-19 at all due to a possibility of faulty tests. The study also only followed the patients for 24 days after their illness, with some scientists questioning whether this was enough time for protective antibodies to be developed. There were also other criticisms about the experimental methods in the study, including that the researchers only looked for antibodies against one viral protein, the spike protein, rather than others which are also important to consider and may also give immunity to SARS-CoV2.
https://www.livescience.com/monkeys-cannot-get-reinfected-with-coronavirus-study.html
https://www.sciencedaily.com/releases/2020/03/200327091234.htm
https://www.scmp.com/news/china/science/article/3078840/coronavirus-low-antibody-levels-raise-questions-about
https://www.medrxiv.org/content/10.1101/2020.03.30.20047365v1
https://www.forbes.com/sites/victoriaforster/2020/04/08/can-you-get-sick-with-coronavirus-again-after-youve-already-had-it-new-research-continues-to-emerge-but-the-jury-is-still-out/#17b2cc5e31a5