Why does Chrome not show the prompt with Allow or Disallow permission to microphone and webcam?

It looks like you may have blocked the domain from ever asking for media access. You can remove this exception from within Chrome Settings.

Open Chrome Settings and then navigate to:

Settings > Privacy (within advanced settings) > Click Content settings.. > Click Manage exceptions...

Make sure your domain does not have Block value for Audio/Video. You can remove the entry for the blocked domain if it exists.

There's a new security feature on Chrome 59:

Temporarily stop permission requests after 3 dismissalsSecurity

Temporarily stop an origin from requesting a permission following the third dismissal of a permission prompt. The stop will be lifted after some amount of time has passed (initially 1 week), after which the origin may request the permission again. A further dismissal will apply the temporary stop again.

For web developers who are testing features, the block can also be removed by changing the permission's setting in the lock icon's page info dialog, or by clearing your browsing data.

Source: https://www.chromestatus.com/features/6443143280984064

I had the exact same SimpleWebRTC setup and it wasn't working in both Chrome and Firefox and I clearly had no approvals/dismissals made before (which could be a reason why browsers don't ask) but I was accessing my local dev environment without https and most modern browsers block HTTP-only transfer of image and audio recorded by the device. Just prepending an 'https' did the trick for me.