Eu resolvi o problema original alterando a maneira como eu configuro o array. Então eu consertei o segundo problema fazendo as variáveis que eu estava usando globalmente.
Dim SignedInNames() As String
Dim NumberSignedIn As Integer
Dim Names() As Variant
Private Sub UserForm_Initialize()
NumberSignedIn = 1
ReDim SignedInNames(0 To NumberSignedIn)
Dim Names() As Variant
Names = Range("OFFSET('People Info'!$A$1,1,0,COUNTA('People Info'!$A:$A)-1,1)")
PersonNameComboBox.List = Names
SignedInListBox.List = SignedInNames
End Sub
Private Sub SignInCommandButton_Click()
ReDim Preserve SignedInNames(0 To NumberSignedIn)
SignedInNames(NumberSignedIn) = PersonNameComboBox.Value
NumberSignedIn = NumberSignedIn + 1
SignedInListBox.List = SignedInNames
End Sub