This really depends on the precise terms and conditions of the company STD and LTD policies, and the "collateral source" rule, designed to prevent multiple recoveries for the same injury, in your state.
Generally if you had a private policy you paid for the carrier would NOT recover the benefits you recover in the litigation or by way of settlement, nor would the disability benefits be taxable.
This is one to ask the lawyer handling your case -- s/he'll know right away.