If you don't specify the schema on creation it will create the object in the users default schema.

If the user doesn't have permissions to alter that schema it will fail.

It is always best practice to double qualify all objects dbo.table, that way if you have table in user1 schema, user2 schema and dbo and your default schema is user2, and you do select * from table, you get user2.table where as you may want dbo.table so always best to prefix