As others have replied, simply counting double-quote characters with  tr/// and discriminating based on even/odd count looks like the best way.

But if just gotta have a regex solution, maybe something like this will serve. Note that this code needs Perl version 5.10+ for the  (*SKIP) (*FAIL) verbs; see Special Backtracking Control Verbs in perlre. (Note also that I use  \x22 extensively in place of  " because otherwise Windows command-line escaping becomes too weird.)

c:\@Work\Perl\monks>perl -wMstrict -le
"use 5.010;
 ;;
 for my $string (
   '\"my\" \"dog\"',
   '\"my\" \"dog shepherd\"',
   'my dog',
   'my \"dog shepherd\"',
   '\"my \"dog\"',
   'my \"dog shepherd',
   ) {
   my $paired   = qr{ \x22 [^\x22]* (?: \\. [^\x22]*)* \x22 }xms;
   my $unpaired = qr{ \x22 [^\x22]* \z }xms;
   ;;
   my $orphan = $string =~ m{ $paired (*SKIP) (*FAIL) | $unpaired }xms
+;
   print qq{'$string' }, $orphan ? 'INVALID' : 'valid';
   }
"
'"my" "dog"' valid
'"my" "dog shepherd"' valid
'my dog' valid
'my "dog shepherd"' valid
'"my "dog"' INVALID
'my "dog shepherd' INVALID
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