It depends. It sounds like the way the company is explaining it to you is almost backwards. I think what they mean to say is that the premiums are paid with AFTER-TAX dollars, in other words, that the premiums do not reduce your taxable income. If this is the case, then your STD payments are not taxable at all. The insurance company could opt to send you a W-2, but this would only report the amount you were paid in box 12 with a code of J, and no other information. This type of W-2 cannot be entered into TurboTax; it is like not receiving a W-2 at all.
When I use the phrase "pre-tax", I refer to premiums that do reduce your taxable income. When this is the case, benefits you receive are taxable. This is the case, for example, if the company is paying the benefit in your behalf. But if you are paying for the premiums, you are probably doing so "after-tax": you have already been taxed on the income you use to pay for the premiums.
Hopefully, this is what is happening. You can verify with your payroll to be certain. But if that is correct information, then you do only report what is shown on your W-2. Your STD is nontaxable and nonreportable income on your tax return.
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