Reland CL to implement browser-side logging to WebRtc log.

This reverts commit d95595eb5145e310baf692f359f9488faf73750d, and it fixes the
original bug where |requests_| was being accessed from the UI thread.

BUG=338848,332261

Review URL: https://codereview.chromium.org/140633004

git-svn-id: svn://svn.chromium.org/chrome/trunk/src@247697 0039d316-1c4b-4281-b951-d872f2087c98
SVN changes up to revision 247697