who can get copies of my personal records

Unless you have a discharge date of 1954 or earlier, only the military veteran (yourself), can request a full copy of your military records. Without your authorization, unrelated members of the public can request your file, but the National Personnel Records Center will only give them certain information. If the researcher wants more information, they can file a Freedom of Information Act (FOIA) request for more of the files. For more information on this, see Access to Official Military Personnel Files (OMPF) - for the General Public and Freedom of Information Act (FOIA) & The Privacy Act.

If you have been discharged in 1954 or earlier, the records are archival and open to the public. More information on this can be found here at Official Military Personnel Files (OMPF), Archival Holdings