During 2009, one half of the payments I received had both social security and Medicare taxes deducted, then suddenly I received the total amount without deductions! Which is correct? Do they owe me for those deductions taken?

QUOTE," Should social security and Medicare taxes be deducted from my private STD payments?"---Your STD benefits are not ssubject to FICA Tax. So,you can get refund on FICA withheld from your STD income.
For taxation( for both your Federal and state(if applicable) taxes), If you pay the premium, then disability is not taxable to you; hiowever,if your employer( if applicable) pays it for you, then the income is taxable to you. If both you and your employer pay the premium half and half, then only your employer's portion will be taxable to you.
However, the taxationof your STD income(distributions), in genral, depends on whether the premiums were paid pretax or after-tax basis. Disability income would be taxable if the premiums were pretax deductions. Your STD income would be tax-exempt if the premiums were paid after-tax. Also most states follow federal rules, so that if it were exempt for federal income tax purposes it would be also exempt for state income taxes.