The objective of all phase III trials is to determine the safety of a vaccine (or new drug) and its efficacy. These trials can usually determine common reactions/side effects, but do not do so well with rare, but serious ones. Those depend on post-marketing surveillance (phase IV). As for efficacy: with a vaccine, the trial can determine if someone develops "markers" associated with immunity, such as particular types of antibody; but unless there is a significant amount of the disease among the persons in the control group, the trial cannot determine whether the vaccine protects, or how long it protects, against the disease itself.
I'm not sure what you mean by " Relapse? Convalescent plasma duration? Sensitivity and specificity?" since they are not relevant to the phase III trials of Covid 19 vaccines.
A vaccination is safe to the extent of endorsement and recommendation that is given by regulatory agencies in each country. The endorsement nonetheless has to follow laid down protocols.
One big problem regarding this issue of vaccinations that may greatly complicate their discovery, testing, and administration is that the novel COVID-19 that existed five months ago is not necessarily the corona virus that is spreading throughout the world today. Just today I heard television network news reports in the U.S. which indicate that, although the corona virus is claiming more victims, the particular strain of the virus is different, possibly less devastating for current victims than for previous ones. This has always been a problem generally for scientists performing experiments intended to increase our knowledge of medical sciences designed to serve humanity. For example, human genetics took forty years to begin to develop, following Gregor Mendel's discovery of genetic laws of inheritance. Obviously, there is a lot of difference between doing experimental research on generations of sweet peas and doing research in the field of human genetics. Therefore, the discussion thread question is very challenging, because, as previous replies to it indicate, there is an unfortunate time lag that is created by the fact that the virus can mutate more quickly than scientists can timely perform research to stop its spread. The "good news" is that the testing phase seems to have demonstrated the ability to respond quickly in the face of the phenomenon of asymptomatic individuals who have become infected by COVID-19. At some point, then, let us hope that a strategy will be deployed whereby testing and vaccinating with be coordinated. Mechanistically speaking, I envision a probability based pattern involving x-number of pharmaceutical companies arriving at y-number of vaccines and the only problem will be to match-up the right vaccine with the right variant of the virus.
In the U.S., there has been a lot of concern about the possibility of premature approval of vaccines for SARS-CoV-2 before there is sufficient evidence from clinical trials to ensure that the vaccines are safe and effective by the "usual standards." The fact is that safety and effectiveness are relative, not absolute. I am not aware of any vaccine that is 100% safe, i.e., never has had any associated side-effects, local or systemic, and 100% effective, i.e., no vaccinee when exposed to the wild-type virus has ever shown any signs of infection. These issues and their relation to government regulation are nicely discussed in a recent paper in JAMA by Avorn and Kesselheim.