The effects of the virus at play are irrelevant... long term, short term, sniffles, death. None of that matters to the topic of moral obligation. Only contagion matters, so let's take two opposing mechanistic scenarios:
1 - Reduced infectiousness (what we expect)
Person A getting the vaccine reduces the chance they will spread it to Person B if exposed and/or infected.
2 - Reduced symptoms (counterintuitive)
Person A getting the vaccine still gets infected and can infect Person B, but no longer has symptoms such as fever. In this situation, Person A actually becomes problematic to the un-vaccinated Person B, since it's not obvious to either individual that Person A is infectious.
In the first scenario, your vaccination potentially helps others. In the second, your vaccination potentially hurts others. The mechanism of action can change altruism into unintended harm. This is why the mechanism of vaccination is so important from social perspective.