I too am looking for a solution to this drag and drop issue.... BUT heres a work around no one has mentioned. If you have a screen shot THAT IS NOT ANNOTATED <--- important, you can Command A (SELECT) and Command C (COPY) from skitch and then Command V (PASTE) in iMessage. If the screen shot is annotated, only the annotations get copied, not the whole screenshot with annotations.
UPDATE. For a screen shot that has annotations...switch to the crop tool and just do a Command C (COPY) from skitch and then Command V (PASTE) in iMessage.. No Command A (SELECT) is needed.
Note the key command to select the crop tool OPTION + COMMAND K
-Cheers - Tommy