Declaring a function in ES6?

now if I'm not wrong the correct "transformation" to es6 would be like this

You're wrong.

Arrow functions are a new syntax with different behaviour. They are not a straight up replacement for function declarations and function expressions (both of which still exist in ES6).

But my ESLint tells me that my logMessage is not defined and I get an error in my console, do I miss something? Do I have to declare var, let or const before the logMessage?

Yes. You're assigning something to a variable. You must declare the variable first.

I also want to export this function from file One to file Two

How you define the function has no bearing on your ability to export it.

function logMessage(message) {
    // etc...
}

... is function declaration which is still perfectly valid in es6. You are converting your function declaration into a function expression, which in es5 would look like this...

logMessage = function(message) {
    // etc...
}

... and then into es6 ...

logMessage = message => {
    // etc
}

... so the linting problem is not introduced by es6 syntax, but rather using function expression, assigning to a variable which without var/let/const is a global variable. There is also a difference in the original function declaration would be hoisted, but the function expression form must be declared before it's called. There is also a difference in that es6 arrow functions retain the context of this from the parent scope, so worth noting they are not 100% direct 1 for 1 mappings of each other.

Short answer, yes, variables need to be declared with var/let/const in order to avoid becoming global variables, whether it's a function or not.

let logMessage = message => {
    // etc
}