is there a question in there?

Since you are overwriting the original array with the results, the simplest solution is:

s/_string$// for @list;
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Which mutates the array in-place, thus avoids constructing two lists (in and out of map) as well as 'curing' the perceived problem.

And if you wanted to create a new list from the modified values whilst retaining the original array unmodified, I'd probably do it this way also:

my @list = ...;
my @newList = @list;
s/_string$// for @newList;
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my @newList = @list;
s/_string$// for @newList
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Or just:

s/_string$// for my @newList = @list;
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What should be done instead:

my @list= qw(first_string second_string);
@list = map((s/_string$// and $_), @list);
print @list;
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This should be done only if one wishes to reduce all strings on which no substitution is done to empty strings:

>perl -wMstrict -le
"my @list= qw(first_string  second_string  THIRD  fourth_string);
 @list = map((s/_string$// and $_), @list);
 printf qq{'$_'  } for @list;
"
'first'  'second'  ''  'fourth'
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If this is not the case, BrowserUk has shown some alternatives.

With 5.14+, the  /r modifier can be used with  s/// substitution to achieve non-mutating substitution with return of a copy of the string (although the intermediate array creation by map noted by BrowserUk will, I think, still occur). See  s/// in the Regexp Quote-Like Operators section of perlop:

If the "/r" (non-destructive) option is used [in s///] then it runs the
substitution on a copy of the string and instead of returning
the number of substitutions, it returns the copy whether or not
a substitution occurred. The original string is never changed ...

>perl -wMstrict -le
"my @list= qw(first_string  second_string  THIRD  fourth_string);
 my @newlist = map s/_string$//r, @list;
 printf qq{'$_'  } for @list;
 print '';
 printf qq{'$_'  } for @newlist;
"
'first_string'  'second_string'  'THIRD'  'fourth_string'
'first'  'second'  'THIRD'  'fourth'
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