52 U.S.C. § 21083 has some relevant text (excerpts below). § 10101 (NVReg Act) does not. By my read, neither bars public access to this data. 5 U.S. Code § 552 (FOIA) probably supersedes.
> (1) ... (v)
> Any election official in the State, including any local election official, may obtain immediate electronic access to the information contained in the computerized list.
> (3) Technological security of computerized list
> The appropriate State or local official shall provide adequate technological security measures to prevent the unauthorized access to the computerized list established under this section
How one votes is secret. Whether one votes is typically compiled by the government and made available to political parties and often the general public.
Note that this is used by campaigns prior to the election to be able to target voters that have not yet voted. Here in Washington state, you receive your mail-in ballot 3-4 weeks before Election Day. I usually return mine as soon as I get it. Campaigns can query this data repeatedly up to Election Day. Knowing that I have already sent in my ballot spares me from needless phone and mailed campaign literature and saves campaigns the cost of sending out useless material.
