Could you answer/verify the following:
1. Verify that the the Social Profile plugin for JomSocial (or whatever extension you're trying to import into) is enabled and the Avatar Import setting is enabled.
2. For the user whose avatar is not being imported, check the tmp directory for the scprofile_XXX_pic_tmp.jpg file, where XXX is the Joomla user id. Please tell me A) whether the file is present or not,

if it is present, if the file size is 0 bytes C) if the present and greater than 0 bytes, view the file and see if it's the real avatar D) the JomSocial images directory is writeable.
Get back to me with those answers and we'll go from there.
-Melissa