Rendering a nested Paragraph's field using Twig
For nested / referenced paragraph rendering, it's probably best to use a separate template as you theorized above by stating &quot;Do we need to (as a guess) create a separate template for the nested paragraph template?&quot;
A general rule of thumb for Twig rendering is, if you are starting to dig that far down in a render array in a given template, best to use the next template inline in the chain.
in addition, In Drupal 8 Paragraphs, there is a nice API function for referenced paragraphs called getParentEntity().
  public function getParentEntity();

  /**
   * Set the parent entity of the paragraph.
   *
   * @param \Drupal\Core\Entity\ContentEntityInterface $parent
   *   The parent entity.
   * @param string $parent_field_name
   *   The parent field name.
   *
   * @return $this
   */

I wrote a blog post dedicated to that here: https://www.dannyenglander.com/blog/drupal-8-architecture-how-to-get-a-value-from-a-paragraphs-parent-entity/
The bottom line is, you will look at Twig template debugging and see the referenced paragraph template suggestion in the output. Then you can use that template with some preprocessing to pull any data you need from the parent paragraph entity. A common use case for this might be some kind of setting like a boolean in the parent entity that you need to pass to the referenced paragraph entity / template.