We can simplify this, reducing the code paths for checking a value and returning and code paths with:

if local_site:
    local_sites = [local_site]
elif request is not None and request.user.is_authenticated():
    local_sites = request.user.local_site.all()
else:
    local_sites = []

for local_site in local_sites:
    if self._is_enabled_for_local_site(...):
        return True

return False

I don't like this returning None as an optional, and don't like not having an explicit return. This way leads to confusing problems down the road. This can easily return a boolean explicitly using basically what we had before:

if local_site.extra_data:
    try:
        return local_site.extra_data[self.EXTRA_DATA_KEY][feature_id]
    except KeyError:
        pass

return False

No trailing comma.

Ship It!

Typo in the description: "browing" -> "browsing"

This would be better as:

if getattr(request, 'local_site', None):
    ...

These two if statements should be combined:

if (local_site.extra_data and
    local_site.extra_data.get(...)):

Extra space before "but".

Ship It!