When you insert Images from the Media Library, and set a caption for them, the caption should show up in the Post when Previewed and Published.
The captions do not show up in the post when Previewed and Published.
The caption for the Image inserted from the Media Library shows up in the Editor:

But not in the Preview:

Or in the Published Post:

1) Switch to Media LIbrary and upload a few Images
2) Create a new Post
3) In this post, insert images from the Media Library to the post and add caption. Observe that the caption shows up in the Editor.
4) Hit Preview, then Publish. Notice that the caption does not display in Preview or Published Post.
Tested on iPhone XR / iOS 13.2.3 / WordPress for iOS 13.7 / NOT using block editor
First tested by @melindahelt, and reported by a user in 2370166-zen and 2532609-zen. It could be related to #12582.
While researching this, I found an earlier report from 2019-08-09:
I upload media with titles and captions. I then confirm there's the title and caption for each photo in the WordPress media section within the WordPress app. I go to post a draft in either block or classic mode and the photo has no caption. When I access the photo from within the draft post there's no caption listed in the info fields. When I leave the draft post and go back to Media, that info is indeed there. I'm so confused.
(internal references: 2262379-zen p4a5px-2qu-p2 cc @ChrissiePollock)
EDIT: actually, that's a separate issue, see https://github.com/wordpress-mobile/WordPress-iOS/issues/13063.
Tested and confirmed that captions added to images in the Aztec editor on the iOS app don't show up in previews or the published post viewed on the front end. (46s)
When I add a caption to an image in the Aztec editor, the HTML source looks something like this:
[caption id=""]<img src="https://madefortesting190318.files.wordpress.com/2019/11/img_20191012_104210.jpg" class="size-full">Lots of plums[/caption]

Tested with WPiOS 13.8.0.0 TestFlight beta on iPhone 6S iOS 13.2.3.
When I add a caption to an image using the classic editor on the web, the caption shows up normally and the HTML source looks something like this:
[caption id="attachment_450" align="alignnone" width="3931"]<img class="alignnone size-full wp-image-450" src="https://madefortesting190318.files.wordpress.com/2019/11/img_20191031_175455.jpg" alt="IMG_20191031_175455" width="3931" height="2948" /> Peppers[/caption]

Tested with WPiOS 13.8.0.0 TestFlight beta on iPhone 6S iOS 13.2.3.
The caption shortcode meta data is missing when you add an image from the media library to a post using Aztec on the iOS app.
@mahangu , @designsimply I investigated the issue a bit and it seems like it aztec issue and better be filed under https://github.com/wordpress-mobile/AztecEditor-iOS
The log shows an error message: “Message: Tag figcaption invalid”
cc: @SergioEstevao
Closing in favor of https://github.com/wordpress-mobile/AztecEditor-iOS/issues/1259. Thank you @mahangu and @yaelirub!