In the US, it would be required that the doctors report it to the VAERS database. There is no need to prove any causal relationship, since that is basically impossible anyway. It doesn't really make sense that Denmark would require you to prove it either, since the data gathering is really the only tool to get evidence of causation.
"In the US, it would be required that the doctors report it to the VAERS database."
From my personal experience and what I've heard online, that seems like wishful thinking. I know two people who had bad heart reactions to the vaccine whose doctors insisted it couldn't be from the vaccine and wouldn't speak of it being connected. And it seems like I've heard similar stories online dozens of times. My guess is the VAERS data dramatically undercounts incidents.
The Pfizer whistleblower paper claimed that even in the early days of the vaccine, internal Pfizer incidents weren't even reported. And given how hostile the entire internet is to discussions of vaccine safety, the idea that people are extremely reluctant to report vaccine injuries seems believable.
Anyone can report to VAERS. Evidenced by the reports if you care to peruse them. There are a lot of gems like, "he got the vax and then just died" and "my baby died a few weeks after I received the vaccine" and my favorite, "my cat died a few days after I got vaccinated". Considering the infinitely higher level of publicity that VAERS has received for this vaccine vs all previous vaccines, my guess is that it is over-reported.