Does the command SET STATISTICS...ON turn statistics information on for all queries running throughout the server or just the current connection?
The answer to the question can be found higher up the document tree:



SET statements:


  The Transact-SQL programming language provides several SET statements
  that change the current session handling of specific information. The
  SET statements are grouped into the categories shown in the following
  table.


Which explains that all these set statements are applied to the current session.

With the category for SET STATISTICS IO,TIME ON; being statistics statements