function saySecret() {
  // Some complicated and possibly side effects
  // operations which are
  // not important for sayHi() test
  return 'Few possible options which are not important';
}

function sayHi(password) {
  let sayHiReturnedValue = 'some value';
  if (password) {
    sayHiReturnedValue += ` ${saySecret()}`; // 👈 Calling the non-export function
  }
  return sayHiReturnedValue;
}

module.exports = { sayHi };

const { expect } = require('chai');
const td = require('testdouble');

const anything = td.matchers.anything();

describe('sayHi()', function () {
  beforeEach(() => {
    td.reset();
    delete require.cache[require.resolve('../../moduleA')];

    this.moduleA = require('../../moduleA');
  });

  it('Should return expected value', () => {
    // ❗ What I would like to achieve: ❗
    const saySecret = td.replace(`somehow replace saySecret()`); // 👈
    td.when(saySecret(anything)).thenReturn('controlled value'); // 👈

    const testedValue = this.moduleA.sayHi('something');
    expect(testedValue).to.equal('some value controlled value');
  });
});