(This answer is basically copied from Peter Murray, specifically these two comments.)
The HTML5 spec allows for boolean attributes with a value of an empty string or the attribute name:
If the attribute is present, its value must either be the empty string or a value that is an ASCII case-insensitive match for the attribute's canonical name, with no leading or trailing whitespace.
So either this:
<div itemscope="" itemtype="http://schema.org/LocalBusiness">
or this:
<div itemscope="itemscope" itemtype="http://schema.org/LocalBusiness">
is valid HTML5.
To be sure that Google recognizes itemscope="itemscope"
correctly, he (Peter Murray) created an example page and ran it through Google's rich snippet validator. From the results, you can see that Google picked up the data (an Event item) correctly.