Here is a complete, working example.
Code
USE master
GO
IF EXISTS (SELECT * FROM sys.databases WHERE name = N'A')
DROP DATABASE A
GO
IF EXISTS (SELECT * FROM sys.databases WHERE name = N'B')
DROP DATABASE B
GO
CREATE DATABASE A
GO
CREATE DATABASE B
GO
USE A
GO
CREATE TABLE X (Col1 varchar(10) NOT NULL)
GO
USE B
GO
CREATE TABLE X2 (Col1 varchar(10) NOT NULL)
GO
USE A
GO
CREATE TRIGGER dbo.trX_Insert
ON dbo.X
FOR INSERT
AS BEGIN
INSERT INTO B.dbo.X2 (Col1)
SELECT Col1
FROM Inserted
END
GO
INSERT INTO A.dbo.X (Col1) VALUES ('This')
INSERT INTO A.dbo.X (Col1) VALUES ('That')
GO
SELECT * FROM A.dbo.X
SELECT * FROM B.dbo.X2
GO
Result
Col1
----------
This
That
Col1
----------
This
That