Does the command SET STATISTICS...ON turn statistics information on for all queries running throughout the server or just the current connection?

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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

This relates to "all Transact-SQL statements that execute in the current connection"
"only within the scope of my connection"

NOT across the whole server