Best way to get identity of inserted row in Linked server?

Yet another variation, in case linked user has permission to call procedures on linked server:

DECLARE @ScopeIdentity int
EXEC [linkedServerName].[database].[schema].sp_executesql N'
  INSERT INTO [table] ...
  SELECT @ScopeIdentityOut = SCOPE_IDENTITY()',
  N'@ScopeIdentityOut INT OUTPUT',
  @ScopeIdentityOut = @ScopeIdentity OUTPUT

Updated per comments on 2019-04-29.

You could use the remote side's sp_executesql:

DECLARE @ScopeIdentity TABLE (ID int);
INSERT INTO @ScopeIdentity
EXEC server.master..sp_executesql N'
  INSERT INTO database.schema.table (columns) VALUES (values);
  SELECT SCOPE_IDENTITY()';
SELECT * FROM @ScopeIdentity;

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Alternatively, you could use OPENQUERY:

SELECT *
FROM OPENQUERY(server, '
  INSERT INTO database.schema.table (columns) VALUES (values);
  SELECT SCOPE_IDENTITY() AS ID');