In Scala, all operators are methods. In order to override an existing method (as Int already defines a + method), the only way would be to inherit and override the + method, and then you'd need to operate on the derived type.
As for overloading, you aren't really overloading Int when defining a def + method in the REPL (quite frankly, I'm quite surprised this method compiles with the use of this in the REPL). All you're doing is creating a + method which takes a single argument. In order to create a new overload for Int, you'll need to use the pimp my library pattern, or in Scala >= 2.10 via an implicit class:
scala> implicit class PimpedInt(x: Int) { | def +(i: Int, s: String): Int = { | println(s) | x + i | } | } defined class PimpedInt scala> 1 + (1, "hello") hello res8: Int = 2
+method you've defined corresponds to the method that gets called in1 + 2?