2020:

Chrome 80 has shipped module workers in February 2020 (and Chrome 82 will ship modules for shared workers). Firefox/Safari do not support those features for now: tests

You may want to use import-from-worker lib to do the heavy lifting for you (for now, you need to check the support for modules in workers and do the fallback by yourself).

ES Modules in workers are already available in Chrome, enabling Experimental Web Platform Features, using the proper flag when launching chrome:

chrome.exe --enable-experimental-web-platform-features

This is the syntax you need to use to load the worker script as a module :

new Worker( 'my-worker.js', { type : 'module' } );

This feature has been in development for almost ayear, and should be available soon, without the need of special flags, however, there is no official release date yet.

As of Nov 21 importing modules in workers still seems to be flaky. One solution is to use rollup to generate an IIFE from your worker as follows:

//worker.js
import { MyModule } from 'my-module.js'
onconnect = async (e) => {
    var port = e.ports[0];
    MyModule.func()

    port.onmessage = (e) => {
        port.postMessage("Hi App");
    }
}

//rollup config
export default [
{
        'input': 'worker.js',
        'output': {
            'file': 'dist/worker.js',
            'format': 'iife'
        },
 },
]


//dist/worker.js (rollup output)

(function () {
    'use strict';
    //MyModule code here, generated by rollup
    
    MyModule.func()
    onconnect = async (e) => {
        var port = e.ports[0];
        port.onmessage = (e) => {
            port.postMessage("Hi App");
        };
    };

}());

//main app

const worker = new SharedWorker("/dist/worker.js");
worker.port.onmessage = (e) => {
    console.log('Message received from worker: ' + e.data);
}
worker.port.postMessage("Hi worker");

Essentially rolllup is doing the work that browsers should be doing. This is working well for me. Of course the code size is increased because the module code is getting copied into worker as well. But it is still DRY as the code is being generated by rollup.

for me assigning to self. worked well. I've put import to another js file: abcImported.js

import { a, b, c } from "./abc.js";

export {  a, b, c };

and in the service worker:

self.a = require('abcImported.js').a;
self.b = require('abcImported.js').b;

in this way, it is accessible inside the worker. (tested in chrome)