First, the statistical information contained in the federal census would be generally available--for example, stats on crops and factories and population numbers. Sherman used this material in planning his Atlanta campaign.
Second, the 1860 federal census would have included where an individual was born, in its individual enumeration. As such, it could show whether or not the person was born in the south. This info was supposed to be kept confidential for a period of years--today, 72 years.
Third, the federal census taking would have been on the local--county and enumeration district--level, then sent to DC. Quite often a copy of that enumeration would be kept by the US census taker, or deposited in the county courthouse. As such, at least some census info was there locally to be used, however, legally, by commanders.
Fourth, info from a census conducted by a state government (e.g., the 1855 Kansas census) would be kept by that state government, and I could see the KS government making use of that info.