This is the view I am talking about:

When it appears as a side bar on the right I am not seeing the addition descriptive text.

It's up to the plugins, not the CMS itself what is displayed in the structure view.

Take an example from a site of mine using djangocms-frontend where plugins are fairly identifiable

What I'd love to see is that the available space is better used, especially if the side bar fills the whole viewport. The text shown is shortened too early after three words by the djangocms-text-ckeditor plugin imho:

There should be an entry here forget_short_description no?
https://docs.django-cms.org/en/latest/reference/plugins.html#cms.models.pluginmodel.CMSPlugin

perhaps also here?
https://docs.django-cms.org/en/latest/how_to/custom_plugins.html#how-to-create-plugins

Yes, there should! Would you like to create a Pull Request to add this? I'd really appreciate it!