I'm having this issue also.  Fresh install on new computer.  Chrome version: Version 53.0.2785.116 m (64-bit).  Windows version: Windows 10 Version 1607 (OS Build 14393.187).

I've attached a screenshot of what hangouts looks like as the issue occurs.

Another strange issue that might be related is that in Chrome settings -> Show Advanced Settings -> Privacy -> Content Settings -> Microphone -> Manage Exceptions, the dialog has no way to edit the exceptions like in other Manage Exceptions dialogs (for example, the one under "Unsandboxed plugin access" looks normal, but the "Camera" one looks the same as the Microphone one).  I've attached a screenshot of the problem dialog.

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I found a workaround via opening hangouts for video chat, and accepting the
microphone permission there, then it also applies in the smaller text/call
window.

There are other hangouts bugs with the new Hangouts system.

Issue has not been modified or commented on in the last 365 days, please re-open or file a new bug if this is still an issue.

For more details visit https://www.chromium.org/issue-tracking/autotriage - Your friendly Sheriffbot

I'm still getting this issue where the Chrome extension is requesting microphone privileges when there is an incoming call and there is no pop up to be found to allow these privileges. I opened a Video Call but the microphone works completely fine and does not request for privileges.

I tried removing permissions from the settings in order to have it trigger properly. I was able to get the permissions to trigger properly when making a video call and accepted but when I tested an incoming call, it showed the microphone issue again.

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Sorry deleted one of the screenshots because it had sensitive info. Here it is with blurs

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1) Remove 'https://hangouts.google.com:443' from Microphone Access.
2) Attempt to make an outgoing call.
- Presented with popup + Chrome mic permission request
3) After accepting the "Allow Mic Access" prompt the popup window fails to close and the chrome extension seems to never receive the message it's been granted mic permissions. 

...

Any attempt to access the microphone via hangouts extension results in an empty popup with no Chrome access prompt. (*see Screenshot)

Note: Bug has been present for (guess) 2-3 months on Chrome:Beta (was hoping it would have been patched by now) 

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Looks like this was reported again here 7 days before reviving this old thread: https://bugs.chromium.org/p/chromium/issues/detail?id=797080

That report has a workaround Hangouts extension in a zip file. Unfortunately it shows an annoying "developer extension" warning when opening Chrome and the warning can't be disabled.

Issue discussion in the product forums here: https://productforums.google.com/forum/?#!msg/hangouts/qrQ_IHVHdRE/uWRfyLzNAwAJ

I'm seeing this issue now too. I use Google Fi, but I've lost my phone. I was expecting to rely on this to be able to make calls, but somehow this 2 year old bug persists.