Tried to reproduce the issue on latest chrome version 70.0.3534.0 using Mac 10.13.6.Attaching screen-cast for reference.
Steps: 
---------
1. Launched reported chrome 
2. Enabled Voice Over and Navigated the URL " https://permission.site/ " -> padlock -> changed to Ask and Reloaded. 
As we are observed that voice Over reads the "permission.site wants to" one time only.

NOTE: First time VoiceOver reading " permission.site Window " and second time reading "permission.site wants to"

@Reporter: Could you please check the attached screen cast and please let us know if anything missed from our end.
Thanks.!

Deleted: 878193.mp4 1.8 MB

	878193.mp4 1.8 MB View Download

I think this is the same bug as on Windows with Jaws / NVDA, but I don't know if they both have the same cause, hence I'll keep separate for now.
I believe the cause is that the title of the bubble appears both as the accessible title of the dialog object and also as a label inside the dialog itself.

The following revision refers to this bug:
  https://chromium.googlesource.com/chromium/src.git/+/2e8f4306edf577bde5d5211b63df598dec30f93c

commit 2e8f4306edf577bde5d5211b63df598dec30f93c
Author: James Wallace-Lee <jamwalla@chromium.org>
Date: Mon Sep 17 23:08:15 2018

Surface title of permissions dialog to accessibility only once

PermissionBubbleDialogDelegateView effectively set its title twice for
accessibility - once by implementing GetWindowTitle and once by setting
BubbleFrameView::SetTitleView. This caused the title to be spoken twice
when screenreaders announced the alert. It is fixed by giving the title
Label the accessible attribute kIgnored.

Bug:  878193 
Change-Id: I21df2a335e1cb02c6ef1921f7d37d5897507fe31
Reviewed-on: https://chromium-review.googlesource.com/1228975
Reviewed-by: Adrienne Porter Felt <felt@chromium.org>
Commit-Queue: James Wallace-Lee <jamwalla@chromium.org>
Cr-Commit-Position: refs/heads/master@{#591864}
[modify] https://crrev.com/2e8f4306edf577bde5d5211b63df598dec30f93c/chrome/browser/ui/views/permission_bubble/permission_prompt_impl.cc

Tried to reproduce the issue on Mac 10.13.6 using reported chrome 70.0.3534.0 and latest chrome 71.0.3556.0.Attaching screen cast for reference.
Steps: 
---------
1. Launched reported chrome 
2. Enabled Voice Over and Navigated the URL " https://permission.site/ " -> padlock -> changed to Ask and Reloaded. 
As we are observed that voice Over reads the "permission.site wants to" one time only.

James Wallace-Lee@ - Could you please check the attached screen cast and please let us know if anything missed from our end. Also help in fixing this issue.

Thanks...!

Deleted: 878193.mp4 2.4 MB

	878193.mp4 2.4 MB View Download

On further check, on the latest canary 71.0.3556.0 on Mac OS 10.13.6, hearing 'permission.site wants to' only once which seems to be expected behavior here. Hence adding the verified label.