How was the authority of government officials lost during the Salem Witch Trials?
The government officials in this case did not lose their legal authority or power, but they lost their moral authority and their legitimacy.
Government officials tend to start to lose their moral authority when they start to do things that the general public does not believe in. When governments do this, they lose the faith of the public and, eventually, come to lose their moral authority (and sometimes even their legal authority) over the people.
This is how things happened during the Salem trials. As more respectable people were accused, others started to lose faith in the authorities. When some of those who were executed acted with great dignity as they were being executed, the government lost even more of the people's faith. This eventually led to a complete loss of faith in the tribunal conducting the trials.
So, the government officials lost their authority by going too far and by doing things that too many of their people could not accept.