Because use strict; and no strict 'refs'; are compile-time directives, you can't easily check for them during run-time. One way to find out if a certain scope has strict bits set, is to save $^H to a variable in a BEGIN block:

#!/usr/bin/perl

STRICT: {
    use strict;
    my $strict;
    BEGIN { $strict = $^H }
    strict_bits($strict);
}

NO_STRICT: {
    no strict;
    my $strict;
    BEGIN { $strict = $^H }
    strict_bits($strict);
}

sub strict_bits {
    my $h = shift;

    # from strict.pm
    my %bitmask = (
        refs => 0x00000002,
        subs => 0x00000200,
        vars => 0x00000400
    );

    printf "\$^H = $h = %b\n", $h;
    print "strict '$_' is ", 
        (($h & $bitmask{$_})
          ? 'on'
          : 'off'
        ),
        "\n"
      for sort keys %bitmask;
}

__END__
$^H = 1794 = 11100000010
strict 'refs' is on
strict 'subs' is on
strict 'vars' is on
$^H = 256 = 100000000
strict 'refs' is off
strict 'subs' is off
strict 'vars' is off
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The same idea works for warnings and ${^WARN_BITS}.

My naive understanding is that $^H is attached to a scope (like the "STRICT" block above) during compile-time, in order to make the strict settings work. Apparently you can't access that setting directly during run-time.

Someone with greater B::Deparse-fu will no doubt be able to explain this in more depth.