I'm not sure I understand your question 100%, but I think that what you are observing is a consequence of the fact that each line you type interactively in the debugger is in its own block/scope. Therefore, by using local you basically limit the change to $/ to that line. Watch:

  DB<1> print "<$/>"
<
>
  DB<2> local $/

  DB<3> print "<$/>"
<
>
  DB<4> local $/; print "<$/>"
<>
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Update: see also this node.