First, is the only thing in widget A a drop down that acts upon widget B? If so they really should not be 2 widgets. The thought process I use is that if a widget does not provide an experience by itself, as in as the only widget on a page, then it should not be a widget. One widget should not depend on another widget for all its functionality.
That being said if it s a situation similar to our comment add widget and a comment list there are a couple of things you can do.
If the intention is a page refresh:
2. Querystring. If the data is non-sensitive(like a dropdown value would be) then just use a redirect and adjust the querystring that is read by widget B.
#set($stringResponse = $core_v2_page.AdjustQueryString($url, $queryString))
3. Temp storage. You can pass data in a sort of storage bag if you will using widget storage:
This returns a key to your data:
#set($stringResponse = $core_v2_utility.StoreTemporaryData($data))
Then use that key to get your data:
#set($stringResponse = $core_v2_utility.GetTemporaryData($key))