Internally Vue uses some RegExps to decide what form of event handler you're using.
If it seems to be the name of a method it will call it and pass the event.
If it seems to be inline code it'll just run it. In this case the event object is accessible as $event.
So:
@click="inputHandler($event)"
is roughly equivalent to:
@click="inputHandler"
Strictly speaking these aren't quite the equivalent, for example with component events that emit multiple arguments you'll only get the first one using $event like this.
See https://v2.vuejs.org/v2/guide/events.html#Methods-in-Inline-Handlers
For a deeper understanding see the Vue source code:
https://github.com/vuejs/vue/blob/0baa129d4cad44cf1847b0eaf07e95d4c71ab494/src/compiler/codegen/events.js#L96
Give your eyes a few minutes to adjust and it isn't too difficult to understand.
Personally I try to avoid anything more complicated than a single method call in the inline listener. Instead I'd suggest having something like @click="onSendClick" and let the method onSendClick worry about the details.