One, as part of your application for benefits you probably signed something allowing your insurance company to request and your doctors to release to them your medical records. You are right, your medical records are private and should not be released without your consent, regardless of their nature (psychiatric, cardiac, oncologic, etc.).
Two, your insurance company needs the opportunity to assess whether you are disabled. They can only do that by examining your medical records as part of your application. Without them, what evidence of your disability have you presented to them? They need something to support their decision to grant benefits, otherwise they are only going to deny your claim.
I Agree. Also, without reviewing your medical records they will not know the nature and extent of your disability and when you have recovered.
DISCLAIMER: David J. McCormick is licensed to practice law in the State of Wisconsin and this answer is being provided for informational purposes only because the laws of your jurisdiction may differ. This answer based on general legal principles and is not intended for the purpose of providing specific legal advice or opinions. Under no circumstances does this answer constitute the establishment of an attorney-client relationship.
The answers provided by Mr. Kerr and Mr. McCormick are correct, but I would add that if the short term disability insurance is an employee benefit (that means you purchased insurance through your employer), then your claim will be under the jurisdiction of ERISA, and you should contact a lawyer who handles ERISA claims because insurance companies look for reasons to deny benefits and the right of the insured to recover damages is very limited in ERISA claims. You may run your question by the ERISA attorney you consult with.