At what point does the responsibility of the manufacturer end with regards to FCC/CE testing?
I live in Europe, and I can't comment on FCC regulations.  

About CE certification there seems to be much misunderstanding. The CE mark on a product claims that the product complies to all relevant CE regulations, but it's the manufacturer who decides to use the marking, not some certifying organisation. If I decide to manufacture some product and place the CE marking on it, I'm free to do so. Did my product undergo required testing? No. Am I liable in case of technical problems due to non-compliance (for instance too much EMI)? Yes. A third party (usually a customer) can challenge my CE-compliance claim and then I have to prove the product is in fact CE-compliant.  

(I wasn't sure about this myself, but I was told so by the director of the quality department of our company, who had a seat in several of those CE committees.)