As a patient, do I have a right to know what legal complaints are made against my doctor?

Answers:
Of course. You can look up the doctor's record with the American Medical Association, your state or local health board, and several other ways.
You can also just ask the doctor. ;)

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Yes.
Complaints and/or lawsuits are generally public record anyway. You just have to go look.

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It's public record. You should be able to search for it at a court database, or maybe find it online on the State Medical license board. I know you can for Chiropractors (my profession).