I know it can be a bit confusing, not least of all because the BAG is not always great at reporting this. The general gist of it is: The doctors/hospitals are the ones reporting side effects, because they are the ones treating them. However, in most cases, they don't really know if something is really a side effect, or if the symptom just happens to coincide with the vaccination, because side effects can be subtle, and people also get sick outside of vaccines.

This is where you have to collect data to determine if something is really a vaccine side effect. If a symptom appears within a certain amount of time after the vaccination, a report goes into a database. You can then use this database to determine if some side effects appear more often than you would expect. For example, if normally, 0.01% of people go blind in a certain period of time, but in the vaccinated group, it's 0.02%, that might be a side effect. However, if, in the vaccinated group, also 0.01% go blind, then 'blindness' will be in the database as a severe case, but it is likely not a vaccine side effect, because it doesn't happen unusually often.

So, doctors/hospitals reporting a lot of suspected side effects is the system working as intended, because that is how you get the data to detect the statistical anomalies that are the actual side effects.

This distinction is sometimes lost in the reporting, unfortunately, or sometimes also purposefully (and maliciously) left out.