Board Thread:General Discussion/@comment-6963992-20130306133828/@comment-2182964-20130306140141

It was said that the Asauchi was with the Shinigami constantly so that they "imprinted" their essence on it. So I'd say it's a mixture of the two: when they trained with it they'd be imprinting a part of their soul on it, though the Shinigami would also do the same if it just happened to be carried with them casually.