Neither browser allows sideloaded extensions in its standard build:
Chrome:
> As of Chrome 33, no external installs are allowed from a path to a local .crx on Windows (see Protecting Windows users from malicious extensions). As of Chrome 44, no external installs are allowed from a path to a local .crx on Mac (see Continuing to protect Chrome users from malicious extensions).
https://developer.chrome.com/apps/external_extensions
Firefox:
> What are my options if I want to install unsigned extensions in Firefox?
> The Nightly and Developer Edition versions of Firefox have a preference to disable signature enforcement.