In the case of heart problems, there is some inconclusive evidence that the Covid vaccine very slightly increases myocarditis and pericarditis risk, especially (Maybe exclusively? I'm not clear on that) in men. In extreme cases this could lead to a heart attack. This would make a person an outlier among outliers and would be exceedingly rare, but so far as I understand it, it is possible. (If you Google it you will get an obligatory Healthline report saying there is no evidence of a link, but I don't know how they're getting from A to C there. Vaccines may increase risk of myocarditis. Myocarditis patients may have a tiny risk of heart attack. Healthline seems to infer that the type of myocarditis vaccines could cause is a special type that could not lead to heart attack, which makes no sense and seems like more 'Let's not panic people' reporting.)
I think in this case the problem is not misinformation, exactly, it is that as humans we are just extraordinarily awful at risk assessment. Basing decisions on large scale statistics is just not something we are wired for. If we see one exceedingly rare risk (side effect) listed next to one huge risk (getting Covid), I think the human brain tends to weigh them as more like a 50/50 risk - two things, two risks. Not a one in a million risk vs. a one in a thousand risk. We are not made to really grok numbers that large.